Updated on 2024/02/01

写真a

 
KATORI Makoto
 
Organization
Faculty of Science and Engineering Professor
Other responsible organization
Physics Course of Graduate School of Science and Engineering, Master's Program
Physics Course of Graduate School of Science and Engineering, Doctoral Program
External link

Degree

  • Doctor of Science ( The University of Tokyo )

  • 理学修士 ( 東京大学 )

Education

  • 1988.3
     

    The University of Tokyo   Graduate School, Division of Science   Physics   doctor course   completed

  • 1984.3
     

    The University of Tokyo   Faculty of Science   Department of Physics   graduated

Research History

  • 1999.4 -  

    ~ 中央大学理工学部教授

  • 1999.4 -  

    - Chuo University, Faculty of Science and Engineering, Professor

  • 1993.4 - 1999.3

    Chuo University   Faculty of Science and Engineering

  • 1993.4 - 1999.3

    . Chuo University, Faculty of Science and Engineering, Associate Professor

  • 1992.4 - 1993.3

    Chuo University   Faculty of Science and Engineering

  • 1992.4 - 1993.3

    . Chuo University, Faculty of Science and Engineering, Lecturer

  • 1988.4 - 1992.3

    The University of Tokyo   Faculty of Science

  • 1988.4 - 1992.3

    . The University of Tokyo, Faculty of Science, Research Associate

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Professional Memberships

  • THE JAPANESE SOCIETY FOR MATHEMATICAL BIOLOGY

  • 日本物理学会

  • 日本数学会

  • The Physical Society of Japan

  • The Mathematical Society of Japan

Research Interests

  • Stochastic Analysis

  • Random Matrix Theory

  • 確率論

  • Phase Transitions and Critical Phenomena

  • 統計力学

  • 数理物理学

  • Probability Theory

  • Statistical Physics

  • Mathematical Physics

Research Areas

  • Natural Science / Mathematical physics and fundamental theory of condensed matter physics

  • Natural Science / Basic analysis

Papers

  • Three-Parametric Marcenko--Pastur Density Reviewed International journal

    Taiki Endo, Makoto Katori

    Journal of Statistical Physics   178 ( 6 )   1397 - 1416   2020.2

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    Authorship:Last author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer  

    The complex Wishart ensemble is the statistical ensemble
    of $M \times N$ complex random matrices with $M \geq N$
    such that the real and imaginary parts of each element
    are given by independent standard normal variables.
    The Marcenko--Pastur (MP) density $\rho(x; r), x \geq 0$
    describes the distribution for squares of the
    singular values of the random matrices in this ensemble
    in the scaling limit $N \to \infty$, $M \to \infty$ with
    a fixed rectangularity $r=N/M \in (0, 1]$.
    The dynamical extension of the squared-singular-value distribution
    is realized by the noncolliding squared Bessel process,
    and its hydrodynamic limit provides the
    two-parametric MP density $\rho(x; r, t)$ with time $t \geq 0$,
    whose initial distribution is $\delta(x)$.
    Recently, Blaizot, Nowak, and Warcho{\l}
    studied the time-dependent complex Wishart ensemble
    with an external source and introduced
    the three-parametric MP density
    $\rho(x; r, t, a)$ by analyzing the
    hydrodynamic limit of the process starting from $\delta(x-a), a > 0$.
    In the present paper, we give useful expressions
    for $\rho(x; r, t, a)$ and perform a systematic study
    of dynamic critical phenomena
    observed at the critical time $t_{\rm c}(a)=a$ when $r=1$.
    The universal behavior in the long-term limit
    $t \to \infty$ is also reported.
    It is expected that the present system having
    the three-parametric MP density provides
    a mean-field model for QCD
    showing spontaneous chiral symmetry breaking.

    DOI: 10.1007/s10955-020-02511-5

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  • From random surfaces to random point processes and random curves: GFF, Dyson model, and multiple SLE Invited

    Makoto Katori

    61 ( 3 )   44 - 50   2023.3

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    Authorship:Lead author   Language:Japanese   Publishing type:Part of collection (book)  

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  • Local universality of determinantal point processes on Riemannian manifolds Reviewed International journal

    Makoto Katori, Tomoyuki Shirai

    Proc. Japan Acad. Ser. A Math. Sci.   98 ( 10 )   95 - 100   2022.12

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    Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:The Japan Academy  

    We consider the Laplace-Beltrami operator $\Delta_g$ on
    a smooth, compact Riemannian manifold $(M,g)$
    and the determinantal point process $\xila$ on $M$ associated with the spectral
    projection of $-\Delta_g$ onto the subspace corresponding to the eigenvalues up to
    $\la^2$.
    We show that the pull-back of $\xila$ by the exponential map
    $\exp_p : T_p^*M \to M$ under a suitable scaling converges
    weakly to the universal determinantal point process on
    $T_p^* M$ as $\la \to \infty$.

    DOI: 10.3792/pjaa.98.018

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  • Hyperuniformity of the determinantal point processes associated with the Heisenberg group Invited International journal

    Makoto Katori

    RIMS Kokyuroku   2235 ( 1 )   12 - 29   2022.12

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    Authorship:Lead author   Language:English   Publishing type:Research paper (bulletin of university, research institution)   Publisher:Research Institute for Mathematical Sciences, Kyoto University  

    The Ginibre point process is given by the eigenvalue
    distribution of a non-hermitian complex Gaussian matrix
    in the infinite matrix-size limit.
    This is a determinantal point process (DPP) on the complex
    plane $\C$ in the sense that all correlation functions
    are given by determinants specified by an integral
    kernel called the correlation kernel.
    Shirai introduced the one-parameter
    ($m \in \N_0$) extensions of the Ginibre DPP
    and called them
    the Ginibre-type point processes.
    In the present paper we consider a generalization
    of the Ginibre and the Ginibre-type point processes on $\C$
    to the DPPs in the higher-dimensional spaces,
    $\C^D, D=2,3, \dots$, in which
    they are parameterized by a multivariate level
    $m \in \N_0^D$.
    We call the obtained point processes
    the extended Heisenberg family of DPPs,
    since the correlation kernels are generally
    identified with the correlations of two points in
    the space of Heisenberg group expressed by
    the Schr\"{o}dinger representations.
    We prove that all DPPs in this large family are
    in Class I of hyperuniformity.

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  • Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases Reviewed International journal

    Taiki Endo, Makoto Katori, Noriyoshi Sakuma

    Symmetry, Integrability and Geometry: Methods and Applications   049   1 - 22   2022.7

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    Authorship:Corresponding author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:SIGMA (Symmetry, Integrability and Geometry: Methods and Application)  

    We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the R-transforms and the S-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the R-transforms and the S-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.

    DOI: 10.3842/sigma.2022.049

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    Other Link: https://www.emis.de/journals/SIGMA/non-commutative-probability.html

  • Partial isometries, duality, and determinantal point processes Reviewed

    Makoto Katori, Tomoyuki Shirai

    Random Matrices: Theory and Applications   11 ( 03 )   2250025/1 - 2250025/70   2022.7

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    Authorship:Corresponding author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:World Scientific Pub Co Pte Ltd  

    A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space S with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

    DOI: 10.1142/s2010326322500253

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  • Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szeg ˝o Kernels and Permanental-Determinantal Point Processes Reviewed International journal

    Makoto Katori, Tomoyuki Shirai

    Communications in Mathematical Physics   392 ( 3 )   1099 - 1151   2022.5

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    Authorship:Corresponding author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer  

    On an annulus
    ${\mathbb{A } }_q :=\{z \in {\mathbb{C } }: q < |z| < 1\}$
    with a fixed $q \in (0, 1)$,
    we study a Gaussian analytic function (GAF) and
    its zero set which defines a point process on ${\mathbb{A } }_q$
    called the zero point process of the GAF.
    The GAF is defined by the
    i.i.d.~Gaussian Laurent series such that
    the covariance kernel parameterized by $r >0$ is identified with
    the weighted Szeg\H{o} kernel of ${\mathbb{A } }_q$
    with the weight parameter $r$
    studied by Mccullough and Shen.
    The GAF and the zero point process
    are rotationally invariant and have a symmetry
    associated with the $q$-inversion of
    coordinate $z \leftrightarrow q/z$ and
    the parameter change $r \leftrightarrow q^2/r$.
    When $r=q$ they are invariant
    under conformal transformations which preserve ${\mathbb{A } }_q$.
    Conditioning the GAF by adding zeros,
    new GAFs are induced such that
    the covariance kernels are also given by
    the weighted Szeg\H{o} kernel of Mccullough and Shen
    but the weight parameter $r$
    is changed depending on the added zeros.

    We also prove that the
    zero point process of the GAF provides
    a permanental-determinantal point process (PDPP)
    in which each correlation function is expressed by
    a permanent multiplied by a determinant.
    Dependence on $r$ of the unfolded 2-correlation function
    of the PDPP is studied.
    If we take the limit $q \to 0$,
    a simpler but still non-trivial
    PDPP is obtained on the unit disk $\D$.
    We observe that the limit PDPP
    indexed by $r \in (0, \infty)$ can be
    regarded as an interpolation between
    the determinantal point process (DPP)
    on ${\mathbb{D } }$ studied by
    Peres and Vir\'ag ($r \to 0$) and
    that DPP of Peres and Vir\'ag with
    a deterministic zero added at the origin ($r \to \infty$).

    DOI: 10.1007/s00220-022-04365-2

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  • Gaussian free fields coupled with multiple SLEs driven by stochastic log-gases Invited Reviewed International journal

    Makoto Katori, Shinji Koshida

    Advanced Studies in Pure Mathematics   87   315 - 340   2021.11

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    Authorship:Lead author   Language:English   Publishing type:Research paper (international conference proceedings)   Publisher:Mathematical Society of Japan  

    Miller and Sheffield introduced the notion of
    an imaginary surface as an equivalence class
    of pairs of simply connected proper subdomains of
    $\C$ and Gaussian free fields (GFFs)
    on them under the conformal equivalence.
    They considered the situation in which
    the conformal transformations are given by
    a chordal Schramm--Loewner evolution (SLE).
    In the present paper, we construct
    GFF-valued processes on $\H$ (the upper half-plane)
    and $\O$ (the first orthant of $\C$)
    by coupling a GFF with a multiple SLE evolving in time on each domain.
    We prove that a GFF on $\H$ and $\O$ is (locally) coupled
    with a multiple SLE if the multiple SLE is driven
    by a stochastic log-gas called the Dyson model defined
    on $\R$ and the Bru--Wishart process
    defined on $\R_+$, respectively.
    We obtain pairs of time-evolutionary domains
    with multiple-slits and GFF-valued processes.

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  • Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions Reviewed International journal

    Makoto Katori, Shinji Koshida

    Journal of Physics A: Mathematical and Theoretical   54 ( 32 )   325002/1 - 325002/19   2021.6

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    The present paper is concerned with properties of multiple Schramm–
    Loewner evolutions (SLEs) labelled by a parameter κ ∈ (0, 8]. Specifically, we consider
    the solution of the multiple Loewner equation driven by a time change of Dyson’s
    Brownian motions in the non-colliding regime. Although it is often considered that
    several properties of the solution can be studied by means of commutation relations
    of SLEs and the absolute continuity, this method is available only in the case that the
    curves generated by commuting SLEs are separated. Beyond this restriction, it is not
    even obvious that the solution of the multiple Loewner equation generates multiple
    curves. To overcome this difficulty, we employ the coupling of Gaussian free fields and
    multiple SLEs. Consequently, we prove the longstanding conjecture that the solution
    indeed generates multiple continuous curves. Furthermore, these multiple curves are
    (i) simple disjoint curves when κ ∈ (0, 4], (ii) intersecting curves when κ ∈ (4, 8), and
    (iii) space-filling curves when κ = 8.

    DOI: 10.1088/1751-8121/ac0dee

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  • Continuum percolation and stochastic epidemic models on Poisson and Ginibre point processes Reviewed International journal

    Machiko Katori, Makoto Katori

    Physica A: Statistical Mechanics and its Applications   581   126191/1 - 126191/14   2021.6

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Elsevier BV  

    The most studied continuum percolation model in two dimensions is the Boolean model consisting of disks with the same radius whose centers are randomly distributed on the Poisson point process (PPP). We also consider the Boolean percolation model on the Ginibre point process (GPP) which is a typical repelling point process realizing hyperuniformity. We think that the PPP approximates a disordered configuration of individuals, while the GPP does a configuration of citizens adopting a strategy to keep social distancing in a city in order to avoid contagion. We consider the SIR models with contagious infection on supercritical percolation clusters formed on the PPP and the GPP. By numerical simulations, we studied dependence of the percolation phenomena and the infection processes on the PPP- and the GPP-underlying graphs. We show that in a subcritical regime of infection rate the PPP-based models show emergence of infection clusters on clumping of points which is formed by fluctuation of uncorrelated Poissonian statistics. On the other hand, the cumulative numbers of infected individuals in processes are suppressed in the GPP-based models.

    DOI: 10.1016/j.physa.2021.126191

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  • Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes Reviewed International journal

    Takato Matsui, Makoto Katori, Tomoyuki Shirai

    Journal of Physics A: Mathematical and Theoretical   54 ( 16 )   165201/1 - 165201/22   2021.4

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    Authorship:Corresponding author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:IOP Publishing  

    The bulk scaling limit of eigenvalue distribution
    on the complex plane ${\mathbb{C } }$
    of the complex Ginibre random matrices provides
    a determinantal point process (DPP).
    This point process is a typical example of
    disordered hyperuniform system characterized
    by an anomalous suppression of large-scale
    density fluctuations.
    As extensions of the Ginibre DPP, we consider
    a family of DPPs defined on the $D$-dimensional
    complex spaces ${\mathbb{C } }$, $D \in {\mathbb{N } }$, in which
    the Ginibre DPP is realized when $D=1$.
    This one-parameter family ($D \in {\mathbb{N } }$) of DPPs
    is called the Heisenberg family,
    since the correlation kernels are
    identified with the Szeg\H{o} kernels for the
    reduced Heisenberg group.
    For each $D$, using the modified Bessel functions,
    an exact and useful expression
    is shown for the local number variance
    of points included in a ball with radius $R$
    in ${\mathbb{R } }^{2D} \simeq {\mathbb{C } }^D$.
    We prove that any DPP in the Heisenberg family is in the
    hyperuniform state of Class I,
    in the sense that the number variance
    behaves as $R^{2D-1}$ as $R \to \infty$.
    Our exact results provide asymptotic expansions
    of the number variances in large $R$.

    DOI: 10.1088/1751-8121/abecaa

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    Other Link: https://iopscience.iop.org/article/10.1088/1751-8121/abecaa/pdf

  • Conformal welding problem, flow line problem, and multiple Schramm–Loewner evolution Reviewed

    Makoto Katori, Shinji Koshida

    Journal of Mathematical Physics   61 ( 8 )   083301 - 083301   2020.8

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    Authorship:Lead author   Language:English   Publishing type:Research paper (scientific journal)   Publisher:AIP Publishing  

    DOI: 10.1063/1.5145357

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  • Scaling limit for determinantal point processes on spheres Invited Reviewed International journal

    Makoto Katori, Tomoyuki Shirai

    RIMS Kokyuroku Bessatsu   B79   123 - 138   2020.4

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    Authorship:Lead author   Language:English   Publishing type:Research paper (bulletin of university, research institution)   Publisher:Research Institute for Mathematical Sciences, Kyoto University  

    The unitary group with the Haar probability measure
    is called Circular Unitary Ensemble.
    All the eigenvalues lie on the unit
    circle in the complex plane and they can be regarded as a
    determinantal point process on ${\mathbb{S } }^1$.
    It is also known that
    the scaled point processes converge weakly to the determinantal point
    process associated with the so-called sine kernel
    as the size of matrices tends to $\infty$.
    We extend this result to the case of high-dimensional spheres
    and show that the scaling limit processes are determinantal
    point processes associated with the kernels expressed by the
    Bessel functions of the first kind.

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  • Two-Dimensional Elliptic Determinantal Point Processes and Related Systems Reviewed

    KATORI Makoto

    Communications in Mathematical Physics   371 ( 3 )   1283 - 1321   2019.11

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    We introduce new families of determinantal point processes (DPPs)<br />
    on a complex plane ${\mathbb{C } }$, which are classified into seven types<br />
    following the irreducible reduced affine root systems,<br />
    $R_N=A_{N-1}$, $B_N$, $B^{\vee}_N$, $C_N$, $C^{\vee}_N$, $BC_N$, $D_N$,<br />
    $N \in {\mathbb{N } }$. <br />
    Their multivariate probability densities are doubly periodic<br />
    with periods $(L, iW)$, $0 &lt; L, W &lt; \infty$, $i=\sqrt{-1}$.<br />
    The construction is based on the orthogonality relations <br />
    with respect to the double integrals over the fundamental domain,<br />
    $[0, L) \times i [0, W)$, which are proved in this paper <br />
    for the $R_N$-theta functions introduced by Rosengren and Schlosser.<br />
    In the scaling limit $N \to \infty, L \to \infty$<br />
    with constant density<br />
    $\rho=N/(LW)$ and constant $W$, we obtain four types of<br />
    DPPs with an infinite number of<br />
    points on ${\mathbb{C } }$, which have periodicity with period $i W$. <br />
    In the further limit $W \to \infty$ with constant $\rho$,<br />
    they are degenerated into three infinite-dimensional DPPs.<br />
    One of them is uniform on ${\mathbb{C } }$ and equivalent <br />
    with the Ginibre point process<br />
    studied in random matrix theory, while<br />
    other two systems are rotationally symmetric around the origin,<br />
    but non-uniform on ${\mathbb{C } }$.<br />
    We show that the elliptic DPP<br />
    of type $A_{N-1}$ is identified with the particle section,<br />
    obtained by subtracting the background effect, <br />
    of the two-dimensional exactly solvable model<br />
    for one-component plasma studied by Forrester.<br />
    Other two exactly solvable models of one-component plasma <br />
    are constructed associated with the elliptic DPPs <br />
    of types $C_N$ and $D_N$.<br />
    Relationship to the Gaussian free field on a torus<br />
    is discussed for these three exactly solvable plasma models.

    DOI: 10.1007/s00220-019-03351-5

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  • Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal point processes Reviewed

    KATORI Makoto

    Journal of Mathematical Physics   60 ( 1 )   013301/1 - 013301/27   2019.4

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    Rosengren and Schlosser introduced notions of<br />
    $\RN$-theta functions for the seven types of<br />
    irreducible reduced affine root systems,<br />
    $\RN=\AN$, $\BN$, $\BNv$, $\CN$, $\CNv$, $\BCN$, $\DN$, $N \in \N$,<br />
    and gave the Macdonald denominator formulas.<br />
    We prove that, if the variables of the<br />
    $\RN$-theta functions are properly scaled<br />
    with $N$, they construct seven sets of biorthogonal <br />
    functions, each of which has a continuous parameter<br />
    $t \in (0, t_{\ast})$ with given $0&lt; t_{\ast} &lt; \infty$.<br />
    Following the standard method in random matrix theory,<br />
    we introduce seven types of one-parameter<br />
    ($t \in (0, t_{\ast})$) families of determinantal point processes<br />
    in one dimension,<br />
    in which the correlation kernels are expressed by<br />
    the biorthogonal theta functions.<br />
    We demonstrate that they are elliptic extensions of<br />
    the classical determinantal point processes<br />
    whose correlation kernels are expressed by<br />
    trigonometric and rational functions.<br />
    In the scaling limits associated with $N \to \infty$,<br />
    we obtain four types of elliptic determinantal point<br />
    processes with an infinite number of points<br />
    and parameter $t \in (0, t_{\ast})$.<br />
    We give new expressions for<br />
    the Macdonald denominators using <br />
    the Karlin--McGregor--Lindstr\&quot;om--Gessel--Viennot<br />
    determinants for noncolliding Brownian paths,<br />
    and show the realization of the associated <br />
    elliptic determinantal point processes<br />
    as noncolliding Brownian brides <br />
    with a time duration $t_{\ast}$, which are<br />
    specified by the pinned configurations<br />
    at time $t=0$ and $t=t_{\ast}$.

    DOI: 10.1063/1.5037805

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  • Excursion Processes Associated with Elliptic Combinatorics Reviewed

    Hiroya Baba, Makoto Katori

    Journal of Statistical Physics   171 ( 6 )   1035 - 1066   2018.6

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:Springer New York LLC  

    Researching elliptic analogues for equalities and formulas is a new trend in enumerative combinatorics which has followed the previous trend of studying q-analogues. Recently Schlosser proposed a lattice path model in the square lattice with a family of totally elliptic weight-functions including several complex parameters and discussed an elliptic extension of the binomial theorem. In the present paper, we introduce a family of discrete-time excursion processes on Z starting from the origin and returning to the origin in a given time duration 2T associated with Schlosser’s elliptic combinatorics. The processes are inhomogeneous both in space and time and hence expected to provide new models in non-equilibrium statistical mechanics. By numerical calculation we show that the maximum likelihood trajectories on the spatio-temporal plane of the elliptic excursion processes and of their reduced trigonometric versions are not straight lines in general but are nontrivially curved depending on parameters. We analyze asymptotic probability laws in the long-term limit T→ ∞ for a simplified trigonometric version of excursion process. Emergence of nontrivial curves of trajectories in a large scale of space and time from the elementary elliptic weight-functions exhibits a new aspect of elliptic combinatorics.

    DOI: 10.1007/s10955-018-2045-6

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  • Hydrodynamic Limit of Multiple SLE Reviewed

    Ikkei Hotta, Makoto Katori

    Journal of Statistical Physics   171 ( 1 )   166 - 188   2018.3

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    Recently del Monaco and Schlei{\ss}inger addressed <br />
    an interesting problem whether one can take the limit of<br />
    multiple Schramm--Loewner evolution (SLE) <br />
    as the number of slits $N$ goes to infinity.<br />
    When the $N$ slits grow from points on the real line $\R$<br />
    in a simultaneous way and go to infinity<br />
    within the upper half plane $\H$,<br />
    an ordinary differential equation <br />
    describing time evolution of the conformal map $g_t(z)$<br />
    was derived in the $N \to \infty$ limit, <br />
    which is coupled with a complex Burgers equation<br />
    in the inviscid limit. <br />
    It is well known that the complex Burgers equation<br />
    governs the hydrodynamic limit of the Dyson model<br />
    defined on $\R$ studied in random matrix theory, <br />
    and when all particles start from<br />
    the origin, the solution of this Burgers equation is given<br />
    by the Stieltjes transformation of the measure which<br />
    follows a time-dependent version of Wigner&#039;s semicircle law.<br />
    In the present paper, first we study the hydrodynamic limit<br />
    of the multiple SLE in the case that all <br />
    slits start from the origin. <br />
    We show that the time-dependent version of Wigner&#039;s <br />
    semicircle law determines the time evolution<br />
    of the SLE hull, $K_t \subset \H\cup \R$ ,<br />
    in this hydrodynamic limit. <br />
    Next we consider the situation such that <br />
    a half number of the slits start from $a&gt;0$ and<br />
    another half of slits start from $-a &lt; 0$,<br />
    and determine the multiple SLE in the <br />
    hydrodynamic limit. <br />
    After reporting these exact solutions,<br />
    we will discuss the universal long-term behavior of <br />
    the multiple SLE and its hull $K_t$ in the hydrodynamic limit.

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  • Percolation of optical excitation mediated by near-field interactions Reviewed

    Makoto Naruse, Song-Ju Kim, Taiki Takahashi, Masashi Aono, Kouichi Akahane, Mario D'Acunto, Hirokazu Hori, Lars Thylen, Makoto Katori, Motoichi Ohtsu

    PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS   471 ( 1 )   162 - 168   2017.4

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    Optical excitation transfer in nanostructured matter has been intensively studied in various material systems for versatile applications. Herein, we theoretically and numerically discuss the percolation of optical excitations in randomly organized nanostructures caused by optical near-field interactions governed by Yukawa potential in a two-dimensional stochastic model. The model results demonstrate the appearance of two phases of percolation of optical excitation as a function of the localization degree of near-field interaction. Moreover, it indicates sublinear scaling with percolation distances when the light localization is strong. Furthermore, such a character is maximized at a particular size of environments. The results provide fundamental insights into optical excitation transfer and will facilitate the design and analysis of nanoscale signal-transfer characteristics. (C) 2016 Elsevier B.V. All rights reserved.

    DOI: 10.1016/j.physa.2016.12.019

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  • Elliptic Determinantal Processes and Elliptic Dyson Models Reviewed

    Makoto Katori

    SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS   13 ( 1 )   p.079,pp.1-36   2017

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    We introduce seven families of stochastic systems of interacting particles in onedimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families A(N-1), B-N, C-N and D-N, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.

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  • Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes Reviewed

    Makoto Katori

    JOURNAL OF MATHEMATICAL PHYSICS   57 ( 10 )   p.103302,pp.1-32   2016.10

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    The Bessel process with parameter D &gt; 1 and the Dyson model of interacting Brownian motions with coupling constant beta &gt; 0 are extended to the processes in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi's theta functions. They are called the elliptic Bessel process, eBES((D)), and the elliptic Dyson model, eDYS((beta)), respectively. Both are realized on the circumference of a circle [0,2 pi r) with radius r &gt; 0 as temporally inhomogeneous processes defined in a finite time interval [0, t(*)), t(*) &lt; infinity. Transformations of them to Schrodinger-type equations with time-dependent potentials lead us to proving that eBES((D)) and eDYS((beta)) can be constructed as the time-dependent Girsanov transformations of Brownian motions. In the special cases where D = 3 and beta = 2, observables of the processes are defined and the processes are represented for them using the Brownian paths winding round a circle and pinned at time t(*). We show that eDYS((2)) has the determinantal martingale representation for any observable. Then it is proved that eDYS((2)) is determinantal for all observables for any finite initial configuration without multiple points. Determinantal processes are stochastic integrable systems in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single continuous function called the spatio-temporal correlation kernel. Published by AIP Publishing.

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  • Bosonic and Fermionic Constructions of Two-Dimensional Quantum Walks

    Hoshiya Yushi, Katori Makoto

    Meeting Abstracts of the Physical Society of Japan   71   2790 - 2790   2016

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    DOI: 10.11316/jpsgaiyo.71.2.0_2790

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  • Phase Diagram of Collective Motion of Bacterial Cells in a Shallow Circular Pool Reviewed

    Jun-ichi Wakita, Shota Tsukamoto, Ken Yamamoto, Makoto Katori, Yasuyuki Yamada

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   84 ( 12 )   p.124001,pp.1-6   2015.12

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    The collective motion of bacterial cells in a shallow circular pool is systematically studied using the bacterial species Bacillus subtilis. The ratio of cell length to pool diameter (i.e., the reduced cell length) ranges from 0.06 to 0.43 in our experiments. Bacterial cells in a circular pool show various types of collective motion depending on the cell density in the pool and the reduced cell length. The motion is classified into six types, which we call random motion, turbulent motion, one-way rotational motion, two-way rotational motion, random oscillatory motion, and ordered oscillatory motion. Two critical values of reduced cell lengths are evaluated, at which drastic changes in collective motion are induced. A phase diagram is proposed in which the six phases are arranged.

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  • Characteristic polynomials of random matrices and noncolliding diffusion processes Invited Reviewed

    Makoto Katori

    RIMS Kokyuroku   1970   22 - 44   2015.11

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  • Self-Elongation with Sequential Folding of a Filament of Bacterial Cells Reviewed

    Ryojiro Honda, Jun-Ichi Wakita, Makoto Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   84 ( 11 )   p.114002,pp.1-7   2015.11

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    Under hard-agar and nutrient-rich conditions, a cell of Bacillus subtilis grows as a single filament owing to the failure of cell separation after each growth and division cycle. The self-elongating filament of cells shows sequential folding processes, and multifold structures extend over an agar plate. We report that the growth process from the exponential phase to the stationary phase is well described by the time evolution of fractal dimensions of the filament configuration. We propose a method of characterizing filament configurations using a set of lengths of multifold parts of a filament. Systems of differential equations are introduced to describe the folding processes that create multifold structures in the early stage of the growth process. We show that the fitting of experimental data to the solutions of equations is excellent, and the parameters involved in our model systems are determined.

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  • Stochastic model showing a transition to self-controlled particle-deposition state induced by optical near-fields Reviewed

    Kan Takahashi, Makoto Katori, Makoto Naruse, Motoichi Ohtsu

    APPLIED PHYSICS B-LASERS AND OPTICS   120 ( 2 )   247 - 254   2015.8

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    We study a stochastic model for the self-controlled particle-deposition process induced by optical near-fields. This process was experimentally realized by Yukutake et al. on an electrode of a novel photovoltaic device as Ag deposition under light illumination, in which the wavelength of incident light is longer than the long-wavelength cutoff of the materials composing the device. Naruse et al. introduced a stochastic cellular automaton model to simulate underlying nonequilibrium processes which are necessary to formulate unique granular Ag film in this deposition process. In the present paper, we generalize their model and clarify the essential role of optical near-fields generated on the electrode surface. We introduce a parameter b indicating the incident light power per site and a function representing the resonance effect of optical near-fields depending on the Ag-cluster size on the surface. Numerical simulation shows a transition from a trivial particle-deposition state to a nontrivial self-controlled particle-deposition state at a critical value b (c), and only in the latter state optical near-fields are effectively generated. The properties of transition in this mesoscopic surface model in nonequilibrium are studied by the analogy of equilibrium phase transitions associated with critical phenomena, and the criteria of transition are reported.

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  • Elliptic determinantal process of type A Reviewed

    Makoto Katori

    PROBABILITY THEORY AND RELATED FIELDS   162 ( 3-4 )   637 - 677   2015.8

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    We introduce an elliptic extension of Dyson's Brownian motion model, which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle. Using elliptic determinant evaluations related to the reduced affine root system of type , we give determinantal martingale representation (DMR) for the process, when it is started at the configuration with equidistant spacing on the circle. DMR proves that the process is determinantal and the spatio-temporal correlation kernel is determined. By taking temporally homogeneous limits of the present elliptic determinantal process, trigonometric and hyperbolic versions of noncolliding diffusion processes are studied.

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  • Determinantal Martingales and Correlations of Noncolliding Random Walks Reviewed

    Makoto Katori

    JOURNAL OF STATISTICAL PHYSICS   159 ( 1 )   21 - 42   2015.4

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    We study the noncolliding random walk (RW), which is a particle system of one-dimensional, simple and symmetric RWs starting from distinct even sites and conditioned never to collide with each other. When the number of particles is finite, , this discrete process is constructed as an -transform of absorbing RW in the -dimensional Weyl chamber. We consider Fujita's polynomial martingales of RW with time-dependent coefficients and express them by introducing a complex Markov process. It is a complexification of RW, in which independent increments of its imaginary part are in the hyperbolic secant distribution, and it gives a discrete-time conformal martingale. The -transform is represented by a determinant of the matrix, whose entries are all polynomial martingales. From this determinantal-martingale representation (DMR) of the process, we prove that the noncolliding RW is determinantal for any initial configuration with , and determine the correlation kernel as a function of initial configuration. We show that noncolliding RWs started at infinite-particle configurations having equidistant spacing are well-defined as determinantal processes and give DMRs for them. Tracing the relaxation phenomena shown by these infinite-particle systems, we obtain a family of equilibrium processes parameterized by particle density, which are determinantal with the discrete analogues of the extended sine-kernel of Dyson's Brownian motion model with . Following Donsker's invariance principle, convergence of noncolliding RWs to the Dyson model is also discussed.

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  • 18aCR-3 Stochastic model of photon breeding process using dressed photon

    Takahashi K., Katori M., Naruse M., Kawazoe T., Ohtsu M.

    Meeting Abstracts of the Physical Society of Japan   70   2755 - 2755   2015

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    DOI: 10.11316/jpsgaiyo.70.2.0_2755

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  • Mathematical aspects of the abelian sandpile models

    Makoto Katori

    Meeting Abstracts of the Physical Society of Japan   70   2839 - 2840   2015

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    DOI: 10.11316/jpsgaiyo.70.2.0_2839

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  • 19pCW-7 Analysis of initial growth process of bacteria by nonlinear systems of differential equations

    Honda R., Wakita J., Katori M.

    Meeting Abstracts of the Physical Society of Japan   70   2866 - 2866   2015

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    DOI: 10.11316/jpsgaiyo.70.2.0_2866

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  • 19pCW-8 Phase diagram of collective motions for bacterial cells in a shallow circular pool

    Wakita Jun-ichi, Tsukamoto Shota, Yamamoto Ken, Katori Makoto

    Meeting Abstracts of the Physical Society of Japan   70   2867 - 2867   2015

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    DOI: 10.11316/jpsgaiyo.70.2.0_2867

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  • 24pAJ-4 Stochastic model showing a transition to self-controlled particle-deposition state induced by optical near-fields

    Takahashi K., Katori M., Naruse M., Ohtsu M.

    Meeting Abstracts of the Physical Society of Japan   70   3214 - 3214   2015

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    DOI: 10.11316/jpsgaiyo.70.1.0_3214

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  • Determinantal martingales and noncolliding diffusion processes Reviewed

    Makoto Katori

    STOCHASTIC PROCESSES AND THEIR APPLICATIONS   124 ( 11 )   3724 - 3768   2014.11

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    Two aspects of noncolliding diffusion processes have been extensively studied. One of them is the fact that they are realized as harmonic Doob transforms of absorbing particle systems in the Weyl chambers. Another aspect is integrability in the sense that any spatio-temporal correlation function can be expressed by a determinant. The purpose of the present paper is to clarify the connection between these two aspects. We introduce a notion of determinantal martingale and prove that, if the system has determinantal-martingale representation, then it is determinantal. In order to demonstrate the direct connection between the two aspects, we study three processes. (C) 2014 Elsevier B.V. All rights reserved.

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  • Oscillatory matrix model in Chern-Simons theory and Jacobi-theta determinantal point process Reviewed

    Yuta Takahashi, Makoto Katori

    JOURNAL OF MATHEMATICAL PHYSICS   55 ( 9 )   p.093302,pp.1-24   2014.9

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    The partition function of the Chern-Simons theory on the three-sphere with the unitary group U(N) provides a one-matrix model. The corresponding N-particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as q-extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the N-particle system to an infinite-dimensional determinantal point process in N -&gt; infinity, in which the correlation kernel is expressed by Jacobi's theta functions. We show that the matrix model obtained by this limit realizes the oscillatory matrix model in Chern-Simons theory discussed by de Haro and Tierz. (C) 2014 AIP Publishing LLC.

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  • Two limiting regimes of interacting Bessel processes Reviewed

    Sergio Andraus, Makoto Katori, Seiji Miyashita

    Journal of Physics A: Mathematical and Theoretical   47 ( 23 )   p.235201,pp.1-30   2014

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    We consider the interacting Bessel processes, a family of multiple-particle systems in one dimension where particles evolve as individual Bessel processes and repel each other via a log-potential. We consider two limiting regimes for this family on its two main parameters: the inverse temperature β and the Bessel index ν. We obtain the time-scaled steady-state distributions of the processes for the cases where β or ν are large but finite. In particular, for largeβ we show that the steady-state distribution of the system corresponds to the eigenvalue distribution of the β-Laguerre ensembles of random matrices. We also estimate the relaxation time to the steady state in both cases. We find that in the freezing regime β → ∞, the scaled final positions of the particles are locked at the square root of the zeroes of the Laguerre polynomial of parameter ν - 1/2 for any initial configuration, while in the regime ν → ∞, we prove that the scaled final positions of the particles converge to a single point. In order to obtain our results, we use the theory of Dunkl operators, in particular the intertwining operator of type B. We derive a previously unknown expression for this operator and study its behaviour in both limiting regimes. By using these limiting forms of the intertwining operator, we derive the steady-state distributions, the estimations of the relaxation times and the limiting behaviour of the processes.

    DOI: 10.1088/1751-8113/47/23/235201

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  • Complex Brownian motion representation of the Dyson model Reviewed

    Makoto Katori, Hideki Tanemura

    Electronic Communications in Probability   18 ( 4 )   1 - 16   2013.1

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    Dyson's Brownian motion model with the parameter β = 2, which we simply call the Dyson model in the present paper, is realized as an h-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of particles, we define a set of entire functions and introduce a martingale for a system of independent complex Brownian motions (CBMs), which is expressed by a determinant of a matrix with elements given by the conformal transformations of CBMs by the entire functions. We prove that the Dyson model can be represented by the system of independent CBMs weighted by this determinantal martingale. From this CBM representation, the Eynard-Mehta-type correlation kernel is derived and the Dyson model is shown to be determinantal. The CBM representation is a useful extension of h-transform, since it works also in infinite particle systems. Using this representation, we prove the tightness of a series of processes, which converges to the Dyson model with an infinite number of particles, and the noncolliding property of the limit process.

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  • System of Complex Brownian Motions Associated with the O'Connell Process Reviewed

    Makoto Katori

    JOURNAL OF STATISTICAL PHYSICS   149 ( 3 )   411 - 431   2012.11

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    The O'Connell process is a softened version (a geometric lifting with a parameter a &gt; 0) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length a. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is N, the rank of the matrix of the Fredholm determinant is N. Then we give a representation for the quantity by using an N-particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit a -&gt; 0 it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered.

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  • Interacting particles on the line and Dunkl intertwining operator of type A: application to the freezing regime Reviewed

    Sergio Andraus, Makoto Katori, Seiji Miyashita

    JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL   45 ( 39 )   p.395201,pp.1-26   2012.10

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    We consider a one-dimensional system of Brownian particles that repel each other through a logarithmic potential. We study two formulations for the system and the relation between them. The first, Dyson's Brownian motion model, has an interaction coupling constant determined by the parameter beta &gt; 0. When beta = 1, 2 and 4, this model can be regarded as a stochastic realization of the eigenvalue statistics of Gaussian random matrices. The second system comes from Dunkl processes, which are defined using differential-difference operators (Dunkl operators) associated with finite abstract vector sets called root systems. When the type-A root system is specified, Dunkl processes constitute a one-parameter system similar to Dyson's model, with the difference that its particles interchange positions spontaneously. We prove that the type-A Dunkl processes with parameter k &gt; 0 starting from any symmetric initial configuration are equivalent to Dyson's model with the parameter beta = 2k. We focus on the intertwining operators, since they play a central role in the mathematical theory of Dunkl operators, but their general closed form is not yet known. Using the equivalence between symmetric Dunkl processes and Dyson's model, we extract the effect of the intertwining operator of type A on symmetric polynomials from these processes' transition probability densities. In the strong coupling limit, the intertwining operator maps all symmetric polynomials onto a function of the sum of their variables. In this limit, Dyson's model freezes, and it becomes a deterministic process with a final configuration proportional to the roots of the Hermite polynomials multiplied by the square root of the process time, while being independent of the initial configuration.

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  • Noncolliding Brownian motion with drift and time-dependent Stieltjes-Wigert determinantal point process Reviewed

    Yuta Takahashi, Makoto Katori

    JOURNAL OF MATHEMATICAL PHYSICS   53 ( 10 )   p.103305   2012.10

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    Using the determinantal formula of Biane, Bougerol, and O'Connell, we give multi-time joint probability densities to the noncolliding Brownian motion with drill, where the number of particles is finite. We study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well-ordered with a scale sigma. We show that, at each time t &gt; 0, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz. Here, one-parameter extensions (theta-extensions) of the Stieltjes-Wigert polynomials, which are themselves q-extensions of the Hermite polynomials, play an essential role. The two parameters a and sigma of the process combined with time t are mapped to the parameters q and theta of the biorthogonal polynomials. By the transformation of normalization factor of our probability density, the partition function of the Chern-Simons matrix model is readily obtained. We study the determinantal structure of the matrix model and prove that, at each time t &gt; 0, the present noncolliding Brownian motion with drill is a determinantal point process, in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel. Using the obtained correlation kernel, we study time evolution of the noncolliding Brownian motion with drift. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4758795]

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  • Reciprocal Time Relation of Noncolliding Brownian Motion with Drift Reviewed

    Makoto Katori

    JOURNAL OF STATISTICAL PHYSICS   148 ( 1 )   38 - 52   2012.7

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    We consider an N-particle system of noncolliding Brownian motion starting from x (1)a parts per thousand currency signx (2)a parts per thousand currency signaEuro broken vertical bar a parts per thousand currency signx (N) with drift coefficients nu (j) , 1a parts per thousand currency signja parts per thousand currency signN satisfying nu (1)a parts per thousand currency sign nu (2)a parts per thousand currency signaEuro broken vertical bar a parts per thousand currency sign nu (N) . When all of the initial points are degenerated to be zero, x (j) =0, 1a parts per thousand currency signja parts per thousand currency signN, the equivalence is proved between a dilatation with factor 1/t of this drifted process and the noncolliding Brownian motion starting from nu (1)a parts per thousand currency sign nu (2)a parts per thousand currency signaEuro broken vertical bar a parts per thousand currency sign nu (N) without drift observed at reciprocal time 1/t, for arbitrary t &gt; 0. Using this reciprocal time relation, we study the determinantal property of the noncolliding Brownian motion with drift having finite and infinite numbers of particles.

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  • Survival Probability of Mutually Killing Brownian Motions and the O'Connell Process Reviewed

    Makoto Katori

    JOURNAL OF STATISTICAL PHYSICS   147 ( 1 )   206 - 223   2012.4

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    Recently O'Connell introduced an interacting diffusive particle system in order to study a directed polymer model in 1+1 dimensions. The infinitesimal generator of the process is a harmonic transform of the quantum Toda-lattice Hamiltonian by the Whittaker function. As a physical interpretation of this construction, we show that the O'Connell process without drift is realized as a system of mutually killing Brownian motions conditioned that all particles survive forever. When the characteristic length of interaction killing other particles goes to zero, the process is reduced to the noncolliding Brownian motion (the Dyson model).

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  • Determinantal Process Starting from an Orthogonal Symmetry is a Pfaffian Process Reviewed

    Makoto Katori

    JOURNAL OF STATISTICAL PHYSICS   146 ( 2 )   249 - 263   2012.1

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    When the number of particles N is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index nu &gt;-1 (BESQ((nu))) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The 2x2 skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, N delta (0), and by the equivalence between the noncolliding BESQ((nu)) and that of the noncolliding squared generalized meander starting from N delta (0).

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  • Noncolliding processes, matrix-valued processes and determinantal processes Invited Reviewed

    Makoto Katori, Hideki Tanemura

    Sugaku Expositions   24 ( 2 )   263 - 289   2011.12

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  • O'Connell's process as a vicious Brownian motion Reviewed

    Makoto Katori

    PHYSICAL REVIEW E   84 ( 6 )   p.061144,pp.1-11   2011.12

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    Vicious Brownian motion is a diffusion scaling limit of Fisher's vicious walk model, which is a system of Brownian particles in one dimension such that if two motions meet they kill each other. We consider the vicious Brownian motions conditioned never to collide with each other and call it noncolliding Brownian motion. This conditional diffusion process is equivalent to the eigenvalue process of the Hermitian-matrix-valued Brownian motion studied by Dyson [J. Math. Phys. 3, 1191 (1962)]. Recently, O'Connell [Ann. Probab. (to be published)] introduced a generalization of the noncolliding Brownian motion by using the eigenfunctions (the Whittaker functions) of the quantum Toda lattice in order to analyze a directed polymer model in 1 + 1 dimensions. We consider a system of one-dimensional Brownian motions with a long-ranged killing term as a generalization of the vicious Brownian motion and construct the O'Connell process as a conditional process of the killing Brownian motions to survive forever.

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  • Markov property of determinantal processes with extended sine, Airy, and Bessel kernels Reviewed

    M. Katori, H. Tanemura

    Markov Processes and Related Fields   17 ( 4 )   541 - 580   2011.12

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  • Velocity correlations of a discrete-time totally asymmetric simple-exclusion process in stationary state on a circle Reviewed

    Yasuyuki Yamada, Makoto Katori

    PHYSICAL REVIEW E   84 ( 4 )   p.041141,pp.1-8   2011.10

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    The discrete-time version of totally asymmetric simple-exclusion process (TASEP) on a finite one-dimensional lattice is studied with the periodic boundary condition. Each particle at a site hops to the next site with probability 0 &lt;= p &lt;= 1 if the next site is empty. This condition can be rephrased by the condition that the number n of vacant sites between the particle and the next particle is positive. Then the average velocity is given by a product of the hopping probability p and the probability that n &gt;= 1. By mapping the TASEP to another driven diffusive system called the zero-range process, it is proved that the distribution function of vacant sites in the stationary state is exactly given by a factorized form. We define k-particle velocity correlation function as the expectation value of a product of velocities of k particles in the stationary distribution. It is shown that it does not depend on positions of k particles on a circle but depends only on the number k. We give explicit expressions for all velocity correlation functions using the Gauss hypergeometric functions. Covariance of velocities of two particles is studied in detail, and we show that velocities become independent asymptotically in the thermodynamic limit.

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  • Extreme value distributions of noncolliding diffusion processes Reviewed

    Minami Izumi, Makoto Katori

    RIMS Kokyuroku Bessatsu   B27   45 - 65   2011.8

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  • Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background Reviewed

    Naoki Kobayashi, Yoshihiro Yamazaki, Hiroto Kuninaka, Makoto Katori, Mitsugu Matsushita, Satoki Matsushita, Lung-Yih Chiang

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   80 ( 7 )   p.074003,pp.1-5   2011.7

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    The statistics of isothermal lines and loops of the cosmic microwave background (CMB) radiation on the sky map is studied and the fractal structure is confirmed in the radiation temperature fluctuation. We estimate the fractal exponents, such as the fractal dimension D-e of the entire pattern of isothermal lines, the fractal dimension D-e of a single isothermal line, the exponent zeta in Korcak's law for the size distribution of isothermal loops, the two kind of Hurst exponents, H-e for the profile of the CMB radiation temperature, and H-c for a single isothermal line. We also perform fractal analysis of two artificial sky maps simulated by a standard model in physical cosmology, the WMAP best-fit Lambda cold dark matter (Lambda CDM) model, and by the Gaussian free model of rough surfaces. The temperature fluctuations of the real CMB radiation and in the simulation using the Lambda CDM model are non-Gaussian, in the sense that the displacement of isothermal lines and loops has an antipersistent property indicated by H-e similar or equal to 0.23 &lt; 1/2.

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  • Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts Reviewed

    Makiko Sato, Makoto Katori

    Physical Review E - Statistical, Nonlinear, and Soft Matter Physics   83 ( 4 )   2011.4

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    As an image of the many-to-one map of loop-erasing operation L of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW ζ is given by the total weight of all random walks π that are inverse images of ζ, {π:L(π)=ζ}. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce a nonintersecting system of N-tuples of LEBPs in a domain D in the complex plane, where the total weight of nonintersecting LEBPs is given by Fomin's determinant of an N×N matrix whose entries are boundary Poisson kernels in D. We set a sequence of chambers in a planar domain and observe the first passage points at which N Brownian paths (γ1,⋯,γN) first enter each chamber, under the condition that the loop-erased parts [L(γ1),⋯,L(γN)] make a system of nonintersecting LEBPs in the domain in the sense of Fomin. We prove that the correlation functions of first passage points of the Brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel. The correlation kernel is of Eynard-Mehta type, which has appeared in two-matrix models and time-dependent matrix models studied in random matrix theory. Conformal covariance of correlation functions is demonstrated. © 2011 American Physical Society.

    DOI: 10.1103/PhysRevE.83.041127

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  • Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts Reviewed

    Makiko Sato, Makoto Katori

    PHYSICAL REVIEW E   83 ( 4 )   p.041127,pp.1-12   2011.4

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    As an image of the many-to-one map of loop-erasing operation L of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW zeta is given by the total weight of all random walks pi that are inverse images of zeta, {pi: L(pi) = zeta}. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce a nonintersecting system of N-tuples of LEBPs in a domain D in the complex plane, where the total weight of nonintersecting LEBPs is given by Fomin's determinant of an N x N matrix whose entries are boundary Poisson kernels in D. We set a sequence of chambers in a planar domain and observe the first passage points at which N Brownian paths (gamma(1), ... , gamma(N)) first enter each chamber, under the condition that the loop-erased parts [L(gamma(1)), ... , L(gamma(N))] make a system of nonintersecting LEBPs in the domain in the sense of Fomin. We prove that the correlation functions of first passage points of the Brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel. The correlation kernel is of Eynard-Mehta type, which has appeared in two-matrix models and time-dependent matrix models studied in random matrix theory. Conformal covariance of correlation functions is demonstrated.

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  • Noncolliding Squared Bessel Processes Reviewed

    Makoto Katori, Hideki Tanemura

    JOURNAL OF STATISTICAL PHYSICS   142 ( 3 )   592 - 615   2011.2

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    We consider a particle system of the squared Bessel processes with index nu &gt;-1 conditioned never to collide with each other, in which if -1 &lt;nu &lt; 0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J (nu) is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.

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  • Dirac equation with an ultraviolet cutoff and a quantum walk Reviewed

    Fumihito Sato, Makoto Katori

    Physical Review A - Atomic, Molecular, and Optical Physics   81 ( 1 )   2010.1

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    The weak convergence theorems of the one- and two-dimensional simple quantum walks, SQW(d),d=1,2, show a striking contrast to the classical counterparts, the simple random walks, SRW(d). In the SRW(d), the distribution of position X(t) of the particle starting from the origin converges to the Gaussian distribution in the diffusion scaling limit, in which the time scale T and spatial scale L both go to infinity as the ratio L/T is kept finite. On the other hand, in the SQW(d), the ratio L/T is kept to define the pseudovelocity V(t)=X(t)/t, and then all joint moments of the components Vj(t),1≤j≤d, of V(t) converge in the T=L→ limit. The limit distributions have novel structures such that they are inverted-bell shaped and their supports are bounded. In the present paper we claim that these properties of the SQW(d) can be explained by the theory of relativistic quantum mechanics. We show that the Dirac equation with a proper ultraviolet cutoff can provide a quantum walk model in three dimensions, where the walker has a four-component qubit. We clarify that the pseudovelocity V(t) of the quantum walker, which solves the Dirac equation, is identified with the relativistic velocity. Since the quantum walker should be a tardyon, not a tachyon,|V(t)|&lt
    c, where c is the speed of light, and this restriction (the causality) is the origin of the finiteness of supports of the limit distributions universally found in quantum walk models. By reducing the number of components of momentum in the Dirac equation, we obtain the limit distributions of pseudovelocities for the lower dimensional quantum walks. We show that the obtained limit distributions for the one- and two-dimensional systems have common features with those of SQW(1) and SQW(2). © 2010 The American Physical Society.

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  • Non-Equilibrium Dynamics of Dyson&apos;s Model with an Infinite Number of Particles Reviewed

    Makoto Katori, Hideki Tanemura

    COMMUNICATIONS IN MATHEMATICAL PHYSICS   293 ( 2 )   469 - 497   2010.1

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    Dyson&apos;s model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant beta/2. We give sufficient conditions for initial configurations so that Dyson&apos;s model with beta = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of Z is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.

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  • Dirac equation with an ultraviolet cutoff and a quantum walk Reviewed

    Fumihito Sato, Makoto Katori

    PHYSICAL REVIEW A   81 ( 1 )   p.012314,pp.1-10   2010.1

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    The weak convergence theorems of the one-and two-dimensional simple quantum walks, SQW((d)), d = 1, 2, show a striking contrast to the classical counterparts, the simple random walks, SRW(d). In the SRW(d), the distribution of position X(t) of the particle starting from the origin converges to the Gaussian distribution in the diffusion scaling limit, in which the time scale T and spatial scale L both go to infinity as the ratio L/root T is kept finite. On the other hand, in the SQW((d)), the ratio L/T is kept to define the pseudovelocity V(t) = X(t)/t, and then all joint moments of the components V-j(t), 1 &lt;= j &lt;= d, of V(t) converge in the T = L -&gt; infinity limit. The limit distributions have novel structures such that they are inverted-bell shaped and their supports are bounded. In the present paper we claim that these properties of the SQW((d)) can be explained by the theory of relativistic quantum mechanics. We show that the Dirac equation with a proper ultraviolet cutoff can provide a quantum walk model in three dimensions, where the walker has a four-component qubit. We clarify that the pseudovelocity V(t) of the quantum walker, which solves the Dirac equation, is identified with the relativistic velocity. Since the quantum walker should be a tardyon, not a tachyon, vertical bar V(t)vertical bar &lt; c, where c is the speed of light, and this restriction (the causality) is the origin of the finiteness of supports of the limit distributions universally found in quantum walk models. By reducing the number of components of momentum in the Dirac equation, we obtain the limit distributions of pseudovelocities for the lower dimensional quantum walks. We show that the obtained limit distributions for the one- and two-dimensional systems have common features with those of SQW((1)) and SQW((2)).

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  • Zeros of Airy Function and Relaxation Process Reviewed

    Makoto Katori, Hideki Tanemura

    JOURNAL OF STATISTICAL PHYSICS   136 ( 6 )   1177 - 1204   2009.9

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    One-dimensional system of Brownian motions called Dyson&apos;s model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is beta/2 times the inverse of particle distance. When beta = 2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson&apos;s model with beta = 2 and N particles, X(t) = (X(1)(t), ... , X(N)(t)), t. [0, infinity), 2 &lt;= N &lt;= infinity, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function Ai(z) is an entire function with zeros all located on the negative part of the real axis R. We consider Dyson&apos;s model with beta = 2 starting from the first N zeros of Ai(z), 0 &gt; a(1) &gt; ... &gt; a(N), N &gt;= 2. In order to properly control the effect of such initial confinement of particles in the negative region of R, we put the drift term to each Brownian motion, which increases in time as a parabolic function: Y(j) (t) = X(j) (t) + t(2)/4 + {d(1) + Sigma(N)(l=1)(1/al)}t, 1 &lt;= j &lt;= N, where d(1) = Ai&apos;(0)/Ai(0). We show that, as the N -&gt; infinity limit of Y(t) = (Y(1)(t), ..., Y(N)(t)), t is an element of [0, infinity), we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of Ai(z) on the negative R is occupied by one particle, to the stationary state mu(Ai). The stationary state mu(Ai) is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on R and in which the Tracy-Widom distribution describes the rightmost particle position.

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  • 非衝突過程・行列値過程・行列式過程 Invited Reviewed

    香取眞理, 種村秀紀

    数学   61 ( 3 )   225 - 247   2009.7

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  • Maximum distributions of bridges of noncolliding Brownian paths Reviewed

    Naoki Kobayashi, Minami Izumi, Makoto Katori

    PHYSICAL REVIEW E   78 ( 5 )   p.051102,pp.1-15   2008.11

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    One-dimensional Brownian motion starting from the origin at time t=0, conditioned to return to the origin at time t=1 and to stay positive during time interval 0 &lt; t &lt; 1, is called the Bessel bridge with duration 1. We consider an N-particle system of such Bessel bridges conditioned never to collide with each other in 0 &lt; t &lt; 1, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximum values of paths attained in the time interval t is an element of(0,1) are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane. For the outermost path, the distribution function of maximum value is exactly determined for general N. We show that the present N-path system of noncolliding Bessel bridges is realized as the positive-eigenvalue process of the 2Nx2N matrix-valued Brownian bridge in the symmetry class C. Using this fact, computer simulations are performed and numerical results on the N dependence of the maximum-value distributions of the inner paths are reported. The present work demonstrates that the extreme-value problems of noncolliding paths are related to random matrix theory, the representation theory of symmetry, and number theory.

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  • Two Bessel bridges conditioned never to collide, double Dirichlet series, and Jacobi theta function Reviewed

    Makoto Katori, Minami Izumi, Naoki Kobayashi

    JOURNAL OF STATISTICAL PHYSICS   131 ( 6 )   1067 - 1083   2008.6

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    It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.

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  • Limit distributions of two-dimensional quantum walks Reviewed

    Kyohei Watabe, Naoki Kobayashi, Makoto Katori, Norio Konno

    PHYSICAL REVIEW A   77 ( 6 )   2008.6

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    One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit t -&gt;infinity of all joint moments of two components of walker&apos;s pseudovelocity, X(t)/t and Y(t)/t, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer simulations is also shown.

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  • Noncolliding Brownian motion and determinantal processes Invited Reviewed

    Makoto Katori, Hideki Tanemura

    JOURNAL OF STATISTICAL PHYSICS   129 ( 5-6 )   1233 - 1277   2007.12

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    A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

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  • Wigner formula of rotation matrices and quantum walks Reviewed

    Takahiro Miyazaki, Makoto Katori, Norio Konno

    Physical Review A - Atomic, Molecular, and Optical Physics   76 ( 1 )   2007.7

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    Quantization of a random-walk model is performed by giving a qudit (a multicomponent wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of the walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's (2j+1) -dimensional unitary representations of rotations with half-integers j 's are useful to analyze the probability laws of quantum walks. For any value of half-integer j, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if (2j+1) is even, the probability measure of limit distribution is given by a superposition of (2j+1) 2 terms of scaled Konno's density functions, and if (2j+1) is odd, it is a superposition of j terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown. © 2007 The American Physical Society.

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  • Wigner formula of rotation matrices and quantum walks Reviewed

    Takahiro Miyazaki, Makoto Katori, Norio Konno

    PHYSICAL REVIEW A   76 ( 1 )   012332   2007.7

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    Quantization of a random-walk model is performed by giving a qudit (a multicomponent wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of the walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's (2j+1)-dimensional unitary representations of rotations with half-integers j's are useful to analyze the probability laws of quantum walks. For any value of half-integer j, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if (2j+1) is even, the probability measure of limit distribution is given by a superposition of (2j+1)/2 terms of scaled Konno's density functions, and if (2j+1) is odd, it is a superposition of j terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.

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  • Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals Reviewed

    Makoto Katori, Hideki Tanemura

    PROBABILITY THEORY AND RELATED FIELDS   138 ( 1-2 )   113 - 156   2007.5

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    Yor's generalized meander is a temporally inhomogeneous modification of the 2(nu + 1)-dimensional Bessel process with nu &gt; -1, in which the inhomogeneity is indexed by kappa epsilon [0, 2(nu + 1)). We introduce the noncolliding particle systems of the generalized meanders and prove that they are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we show that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions J(nu) used in the fractional calculus, where orders of differintegration are determined by nu - kappa. As special cases of the two parameters (nu, kappa), the present infinite systems include the quaternion determinantal processes studied by Forrester, Nagao and Honner and by Nagao, which exhibit the temporal transitions between the universality classes of random matrix theory.

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  • Quantum walks and orbital states of a Weyl particle (vol 72, pg 012316, 2005) Reviewed

    M Katori, S Fujino, N Konno

    PHYSICAL REVIEW A   72 ( 1 )   12316   2005.7

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  • Quantum walks and orbital states of a Weyl particle Reviewed

    M Katori, S Fujino, N Konno

    PHYSICAL REVIEW A   72 ( 1 )   2005.7

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    The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin 1/2, in which each wave number k of the walker's wave function is mapped to a point q(k) in the three-dimensional momentum space and q(k) makes a planar orbit as k changes its value in [-pi,pi). The integration over k providing the real-space wave function for a quantum walker corresponds to considering an orbital state of a Weyl particle, which is defined as a superposition (curvilinear integration) of the energy-momentum eigenstates of a free Weyl equation along the orbit. Konno's novel distribution function of a quantum walker's pseudovelocities in the long-time limit is fully controlled by the shape of the orbit and how the orbit is embedded in the three- dimensional momentum space. The family of orbital states can be regarded as a geometrical representation of the unitary group U (2) and the present study will propose a new group-theoretical point of view for quantum-walk problems.

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  • Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems Reviewed

    M Katori, H Tanemura

    JOURNAL OF MATHEMATICAL PHYSICS   45 ( 8 )   3058 - 3085   2004.8

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    As an extension of the theory of Dyson's Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor's generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish-Chandra (Itzykson-Zuber) formula of integral over unitary group is established. (C) 2004 American Institute of Physics.

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  • Infinite Systems of Non-Colliding Brownian Particles Invited Reviewed

    Makoto Katori, Taro Nagao, Hideki Tanemura

    Advanced Studies in Pure Mathematics,Stochastic Analysis on Large Scale Interacting Systems   39   283 - 306   2004.2

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  • Dualities for the Domany-Kinzel model Reviewed

    M Katori, N Konno, A Sudbury, H Tanemura

    JOURNAL OF THEORETICAL PROBABILITY   17 ( 1 )   131 - 144   2004.1

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    We study the Domany-Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (p(1), p(2)) is an element of [0, 1](2). When p(1) = alphabeta and p(2) = alpha(2beta - beta(2)) with (alpha, beta) is an element of [0, 1](2), the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities alpha of a site being open and beta of a bond being open. This paper treats dualities for the Domany-Kinzel model xi(t)(A) and the DKdual eta(t)(A) starting from A. We prove that (i) E(x|(xitA boolean AND B)|)= E(x|(xitBboolean ANDA|)) if x = 1-(2p(1) - p(2))/ p(1)(2), (ii) E(x|xi(t)(Aboolean ANDB)|)= E(x|(etatBboolean ANDA|)) if x= 1-(2p(1) - p(2))/p(1), and (iii) E(x(|etatAboolean ANDB|))= E(x|(etatBboolean ANDA|)) if x= 1-(2p(1) - p(2)), as long as one of A, B is finite and p(2) less than or equal to p(1).

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  • Functional central limit theorems for vicious walkers Reviewed

    Makoto Katori, Hideki Tanemura

    Stochastics and Stochastics Reports   75 ( 6 )   369 - 390   2003.12

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  • Noncolliding Brownian motions and Harish-Chandra formula Reviewed

    Makoto Katori, Hideki Tanemura

    Elect. Comm. in Probab.   8   112 - 121   2003.9

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  • Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov-de Gennes random matrices Reviewed

    M Katori, H Tanemura, T Nagao, N Komatsuda

    PHYSICAL REVIEW E   68 ( 2 )   p.021112   2003.8

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    Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov-de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type CI in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.

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  • Moments of vicious walkers and Mobius graph expansions Reviewed

    M Katori, N Komatsuda

    PHYSICAL REVIEW E   67 ( 5 )   p.051110   2003.5

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    A system of Brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval (0,T] is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time t goes on from 0 to T. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit t--&gt;0, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called Mobius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of nonorientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the Mobius expansion using the Stirling numbers of the first kind.

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  • Dynamical correlations among vicious random walkers Reviewed

    T Nagao, M Katori, H Tanemura

    PHYSICS LETTERS A   307 ( 1 )   29 - 35   2003.1

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    Nonintersecting motion of Brownian particles in one dimension is studied. The system is constructed as the diffusion scaling limit of Fisher's vicious random walk. N particles start from the origin at time t = 0 and then undergo mutually avoiding Brownian motion until a finite time t = T. In the short time limit t much less than T, the particle distribution is asymptotically described by Gaussian Unitary Ensemble (GUE) of random matrices. At the end time t = T, it is identical to that of Gaussian Orthogonal Ensemble (GOE). We show that the most general dynamical correlations among arbitrary number of particles at arbitrary number of times are written in the forms of quaternion determinants. Asymptotic forms of the correlations in the limit N --&gt; infinity are evaluated and a discontinuous transition of the universality class from GUE to GOE is observed. (C) 2002 Elsevier Science B.V. All rights reserved.

    DOI: 10.1016/S0375-9601(02)01661-4

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  • Families of vicious walkers Reviewed

    J Cardy, M Katori

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   36 ( 3 )   609 - 629   2003.1

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    We consider a generalization of the vicious walker problem in which N random walkers in R-d are grouped into p families. Using field-theoretic renormalization group methods we calculate the asymptotic behaviour of the probability that no pairs of walkers from different families have met up to time t. For d &gt; 2, this is constant, but for d &lt; 2 it decays as a power t(-alpha), which we compute to O(82) in an expansion in epsilon = 2 - d. The second-order term depends on the ratios of the diffusivities of the different families. In two dimensions, we find a logarithmic decay (ln t)(-(&alpha;) over bar) and compute (&alpha;) over bar exactly.

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  • Moments of vicious walkers and Möbius graph expansions Reviewed

    Makoto Katori, Naoaki Komatsuda

    Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics   67 ( 5 )   10   2003

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    A system of Brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval [Formula presented] is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time t goes on from 0 to T. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit [Formula presented] only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called Möbius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of nonorientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the Möbius expansion using the Stirling numbers of the first kind. © 2003 The American Physical Society.

    DOI: 10.1103/PhysRevE.67.051110

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  • Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov–de Gennes random matrices Reviewed

    Makoto Katori, Hideki Tanemura, Taro Nagao, Naoaki Komatsuda

    Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics   68 ( 2 )   16   2003

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    Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov–de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type [Formula presented] in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified. © 2003 The American Physical Society.

    DOI: 10.1103/PhysRevE.68.021112

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  • Vicious Walker Model, Schur Function and Random Matrices

    Katori Makoto

    Bulletin of the Japan Society for Industrial and Applied Mathematics   13 ( 4 )   296 - 307   2003

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    We consider the vicious walker model, which was introduced by Michael Fisher in his Boltzmann medal lecture in 1983 as a mathematical model of wetting and melting phenomena. It is a system of particles performing noncolliding random walk in one dimension. Using nonintersecting property of the paths of vicious walkers and by elementary calculus of deter minants, we show that the Green function of the system is equal to the Schur function, which plays an important role in the representation theory of symmetric group, and its two kinds of determinantal expressions are derived. MacMahon conjecture, Bender-Knuth conjecture and Macdonald equality for the summations of Schur functions are discussed from the viewpoint of vicious walker model. By taking the diffusion scaling limit of the vicious walker model, a system of noncolliding Brownian particles is constructed and its relation to the distribution of eigenvalues of real symmetric random matrices is clarified.

    DOI: 10.11540/bjsiam.13.4_296

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  • Scaling limit of vicious walks and two-matrix model Reviewed

    Makoto Katori, Hideki Tanemura

    Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics   66 ( 1 )   2002.7

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    We consider the diffusion scaling limit of the one-dimensional vicious walker model of Fisher and derive a system of nonintersecting Brownian motions. The spatial distribution of N particles is studied and it is described by use of the probability density function of eigenvalues of [formula presented] Gaussian random matrices. The particle distribution depends on the ratio of the observation time t and the time interval T in which the nonintersecting condition is imposed. As [formula presented] is going on from 0 to 1, there occurs a transition of distribution, which is identified with the transition observed in the two-matrix model of Pandey and Mehta. Despite of the absence of matrix structure in the original vicious walker model, in the diffusion scaling limit, accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution. © 2002 The American Physical Society.

    DOI: 10.1103/PhysRevE.66.011105

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  • Scaling limit of vicious walks and two-matrix model Reviewed

    M Katori, H Tanemura

    PHYSICAL REVIEW E   66 ( 1 )   p.011105   2002.7

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    We consider the diffusion scaling limit of the one-dimensional vicious walker model of Fisher and derive a system of nonintersecting Brownian motions. The spatial distribution of N particles is studied and it is described by use of the probability density function of eigenvalues of NxN Gaussian random matrices. The particle distribution depends on the ratio of the observation time t and the time interval T in which the nonintersecting condition is imposed. As t/T is going on from 0 to 1, there occurs a transition of distribution, which is identified with the transition observed in the two-matrix model of Pandey and Mehta. Despite of the absence of matrix structure in the original vicious walker model, in the diffusion scaling limit, accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution.

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  • Limit theorems for the nonattractive Domany-Kinzel model Reviewed

    M Katori, N Konno, H Tanemura

    ANNALS OF PROBABILITY   30 ( 2 )   933 - 947   2002.4

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    We study the Domany-Kinzel model, which is a class of discrete time Markov processes with two parameters (p(1), p(2)) is an element of [0, 1](2) and whose states are subsets of Z, the set of integers. When p(1) = alphabeta and p(2) = alpha(2beta - beta(2)) with (alpha, beta) is an element of [0, 1](2), the process can be identified with the mixed site-bond oriented percolation model on a square lattice with the probabilities of open site alpha and of open bond beta. For the attractive case, 0 less than or equal to p(1) less than or equal to p(2) less than or equal to 1, the complete convergence theorem is easily obtained. On the other hand, the case (p(1), p(2)) = (1, 0) realizes the rule 90 cellular automaton of Wolfram in which, starting from the Bernoulli measure with density 0, the distribution converges weakly only if theta is an element of {0, 1/2, 1}. Using our new construction of processes based on signed measures, we prove limit theorems which are also valid for nonattractive cases with (p(1), p(2)) not equal (1, 0). In particular, when p(2) is an element of [0, 1] and p(1) is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.

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  • Analysis of trajectories of random walkers on spatio-temporal plane Reviewed

    N.Inui, M.Katori

    Trans. Mat. Res. Soc. Jpn.   26   409 - 412   2001.3

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  • Trajectories of two friendly walkers Reviewed

    M.Katori, N.Inui

    Trans. Mat. Res. Soc. Jpn.   26   405 - 407   2001.3

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  • Bounds on the critical values of the one-dimensional diffusion contact process Reviewed

    N.Konno, S.Hayashi, A.Sudbury, H.Tanemura, M.Katori

    Trans. Mat. Res. Soc. Jpn.   26   385 - 388   2001.3

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  • Dualities for discrete-time stochastic models in ome dimension Reviewed

    N.Konno, A.Sudbury, H.Tanemura, M.Katori

    Trans. Mat. Res. Soc. Jpn.   26   381 - 384   2001.3

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  • Dependence on direction for spatial distributions for pivotal bonds on percolation \\ Reviewed

    S.Kogure, M.Sekine, M.Katori, N.Konno

    Trans. Mat. Res. Soc. Jpn.   26   369 - 372   2001.3

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  • Analysis of canopy-gap structures of forests by theomal equilibirum correlation equalities Reviewed

    T.Takamatsu, M.Katori

    Trans. Mat. Res. Soc. Jpn.   26   397 - 400   2001.3

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  • On the Drossel-Schwabl limit in a forest fire model Reviewed

    T.Takamatsu, M.Katori

    Trans. Mat. Res. Soc. Jpn.   26   401 - 404   2001.3

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  • Low-density series expansion for the Domany-Kinzel model Reviewed

    N Inui, M Katori, FM Bhatti

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   70 ( 2 )   359 - 366   2001.2

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    Domany-Kinzel (DK) model is a family of the 1+1 dimensional stochastic cellular automata with two parameters p(1) and p(2), which simulate time evolution of interacting active elements in a random medium. By identifying a set of active sites on the spatio-temporal plane with a percolation cluster, we discuss the directed percolation (DP) transitions in the DK model. We parameterize p(1) = p and p(2) = alphap with p is an element of [0, 1] and alpha is an element of [0, 2] and calculate the mean cluster size and other quantities characterizing the DP cluster as the series of p up to order 51 for several values of alpha by using a graphical expansion formula recently given by Konno and Katori. We analyze the series by the first- and second-order differential approximations and the Zinn-Justin method and study the dependence on alpha of the convergence of estimations of critical values and critical exponents. In the mixed site-bond DP region; 1 less than or equal to alpha less than or equal to 1.3553, the convergence is excellent. As alpha -&gt; 2 slowing down of convergence and as alpha -&gt; 0 peculiar oscillation of estimations are observed. This paper is the first report of the systematic study of DK model by series expansion method.

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  • Statistical properties of trajectories of friendly walkers on spatio-temporal plane Reviewed

    N Inui, M Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   70 ( 1 )   78 - 85   2001.1

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    Friendly walkers are the non-crossing random walkers on a lattice with attractive interactions. We characterize each trajectory of friendly walkers by the number of walkers nz, the time interval of observation t and the total length of trajectory r. A new algorithm to generate trajectories on a spatio-temporal plane is proposed and the distribution function of number of distinct trajectories characterized by (m, f, r), f(m,t)(r), is estimated by a random sampling method. The variance, the skewness and the kurtosis of f(m,t)(r) converge to finite values without scaling as t --&gt; infinity for each m. The distribution is asymmetric and its tails are expressed by stretched exponential functions. We consider the canonical distribution of m friendly walkers by introducing a parameter p which plays the same role of the Boltzmann factor e(-beta) in the usual equilibrium systems. We calculate the mean and variance of r in the canonical distribution as a function of p for each m at t. It is observed that the variance of r per unit time interval has a peak at a certain value of p for each m = 2, 3, 4 and 5. We discuss the possibility that the peak indicates the phase transition of trajectories of friendly walkers realized on a spatio-temporal plane.

    DOI: 10.1143/JPSJ.70.78

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  • Fermi partition functions of friendly walkers and pair connectedness of directed percolation Reviewed

    N Inui, M Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   70 ( 1 )   1 - 4   2001.1

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    Non-crossing random walkers with attractive interactions called friendly walkers (FWs) are studied. A restriction on trajectories, which is analogous to Pauli's exclusion principle, is imposed and the Fermi partition functions are defined. We prove a theorem that the pair connectedness of the bond directed percolation (DP) with bond concentration p is related to the Fermi grand partition function of FW if we set, the temperature T = -1/(k(B) ln p) and the chemical potential mu = -i pi /lnp, where k(B) is the Boltzmann constant and i = root -1. The pure imaginary chemical potential means that the DP transition can be regarded as a symmetry breaking of parity in the number of FWs. As a corollary of the theorem, a new method is proposed for calculating the series expansion of the pair connectedness and percolation probability of DP using the low-temperature expansion data of FW.

    DOI: 10.1143/JPSJ.70.1

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  • Extenstion of the Arrowsmith-Essam formula to the Domany-Kinzel model Reviewed

    N.Konno, M.Katori

    J. Stat. Phys   101   747 - 774   2000.10

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  • Exact Results for the Abelian Sandpile Models

    KATORI Makoto

    Butsuri   55 ( 4 )   276 - 280   2000.4

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    DOI: 10.11316/butsuri1946.55.276

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  • Percolation transitions and wetting transitions in stochastic models Invited Reviewed

    M Katori

    BRAZILIAN JOURNAL OF PHYSICS   30 ( 1 )   83 - 96   2000.3

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    Stochastic models with irreversible elementary processes are introduced, and their macroscopic behaviors in the infinite-time and infinite-volume limits are studied extensively, in order to discuss nonequilibrium stationary states and phase transitions. The Domany-Kinzel model is a typical example of such an irreversible particle system. We first review this model, and explain that in a certain parameter region, the nonequilibrium phase transitions it exhibits can be identified with directed percolation transitions on the spatio-temporal plane. We then introduce an interacting particle system with particle conservation called friendly walkers (FW). It is shown that the m = 0 limit of the correlation function of m friendly walkers gives the correlation function of the Domany-Kinzel model, if we choose the parameters appropriately. We show that FW can be considered as a model of interfacial wetting transitions, and that the phase transitions and critical phenomena of FW can be studied using Fisher's theory of phase transitions in linear systems. The FW model may be the key to constructing a unified theory of directed percolation transitions and wetting transitions. Descriptions of FW as a model of interacting vicious walkers and as a vertex model are also given.

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  • Survival probabilities for discrete-time models in one dimension Reviewed

    M.Katori, N.Konno, H.Tanemura

    J. Stat. Phys.   99   603 - 612   2000.2

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    DOI: 10.1023/A:1018617328216

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  • Proof of breaking of self-organized criticality in a nonconservative Abelian sandpile model Reviewed

    T Tsuchiya, M Katori

    PHYSICAL REVIEW E   61 ( 2 )   1183 - 1188   2000.2

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    computer simulations, it was reported that the Bak-Tang-Wiesenfeld (BTW) model loses self-organized criticality (SOC) when some particles are annihilated in a toppling process in the bulk of system. We give a rigorous proof that the BTW model loses SOC as soon as the annihilation rate becomes positive. To prove this, a nonconservative Abelian sandpile model is defined on a square lattice, which has a parameter alpha (greater than or equal to 1) representing the degree of breaking of the conservation law. This model is reduced to be the BTW model when alpha = 1. By calculating the average number of topplings in an avalanche &lt; T &gt; exactly, it is shown that for any alpha &gt; 1, &lt; T &gt; &lt; infinity even in the infinite-volume limit. The power-law divergence of &lt; T &gt; with an exponent 1 as alpha --&gt;1 gives a scaling relation 2 v(2 - a) = 1 for the critical exponents v and a of the distribution function of T. The 1-1 height correlation C-11(r) is also calculated analytically and we show that C-11(r) is bounded by an exponential function when alpha &gt; 1, although C-11(r)similar to r(-2d) was proved by Majumdar and Dhar for the d-dimensional BTW model. A critical exponent v(11) characterizing the divergence of the correlation length xi as alpha --&gt; 1 is defined as xi similar to \alpha - 1\(-v11) and our result gives an upper bound v(11)less than or equal to 1/2.

    DOI: 10.1103/PhysRevE.61.1183

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  • Baxter-Guttmann-Jensen conjecture for power series in directed percolation problem Invited Reviewed

    M.Katori, T.Tsuchiya, N.Inui, H.Kakuno

    Annals of Combinatorics   3   337 - 356   1999.10

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  • Analysis of canopy-gap structures of forests by Ising-Gibbs states - Equilibrium and scaling property of real forests Reviewed

    S Kizaki, M Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   68 ( 8 )   2553 - 2560   1999.8

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    Canopy is the uppermost branchy layer of trees and spatial distribution of canopy gaps is important for the coexistence of species and the forest dynamics. A new method to analyze the canopy-gap structure is proposed, in which configurations of canopy-gap sites are approximated by the Ising-Gibbs states with two parameters. Results of the application to the real data of a neotropical forest in Barro Colorado Island, Panama, and of a deciduous forest in Ogawa Forest Reserve, Japan, show the validity of our method. Canopy-gap structures of the forests are not exactly critical but nearly critical and the scaling argument of the Ising spin configurations is useful to explain the power-law distributions of gap sizes reported by ecologists.

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  • Stochastic lattice model for locust outbreak Reviewed

    Shinya Kizaki, Makoto Katori

    Physica A: Statistical Mechanics and its Applications   266 ( 1-4 )   339 - 342   1999.4

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    The locust is a kind of grasshoppers. Gregarious locusts form swarms and can migrate over large distances and they spread and damage a large area (locust outbreak). When the density is low, each of locusts behaves as an individual insect (solitary phase). As locusts become crowded, they become to act as a part of a group (gregarious phase) as a result of interactions among them. Modeling of this phenomenon is a challenging problem of statistical physics. We introduce a stochastic cellular automaton model of locust population-dynamics on lattices. Change of environmental conditions by seasonal migration is a key factor in gregarisation of locusts and we take it into account by changing the lattice size periodically. We study this model by computer simulations and discuss the locust outbreak as a cooperative phenomena.

    DOI: 10.1016/S0378-4371(98)00613-X

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  • Effect of anisotropy on the self-organized critical states of Abelian sandpile models Reviewed

    Tomoko Tsuchiya, Makoto Katori

    Physica A: Statistical Mechanics and its Applications   266 ( 1-4 )   358 - 361   1999.4

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    The directed Abelian sandpile models are defined on a square lattice by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, in which c = 1 is for the isotropic case. We calculate the expected number of the topplings per added particle, 〈T〉, which depends on the lattice size L as Lx for large L. Our exact solution gives that x = 1 when any anisotropy is included in the system, while x = 2 in the isotropic case. These results allow us to introduce a new critical exponent, θ, defined by χ ≡ 1imL→∞ 〈T〉/L with c ≠ 1 as χ approx. |c - 1|-θ for |c - 1| ≪ 1. From the explicit expression of 〈T〉, we obtain θ = 1.

    DOI: 10.1016/S0378-4371(98)00616-5

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  • Exact results for the directed Abelian sandpile models Reviewed

    T Tsuchiya, M Katori

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   32 ( 9 )   1629 - 1641   1999.3

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    We define the directed Abelian sandpile models by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, where c = 1 is for the isotropic case. We calculate some quantities characterizing the self-organized critical states on the one- and two-dimensional lattices. In particular, we obtain the expected number of topplings per added particle, [T], which shows the dependence on the lattice size L as L-x for large L. We show that the critical exponent x does not depend on the dimensionality d, at least ford = 1 and 2, and that when any anisotropy is included in the system x = 1, while x = 2 in the isotropic system. This result gives a rigorous proof of the conjecture by Kadanoff et al (1989 Phys. Rev. A 39 6524-37) that the anisotropy will distinguish different universality classes. We introduce a new critical exponent, theta, defined by chi = lim(L--&gt;infinity)[T]/L with c not equal 1 as chi similar to \c - 1\(-theta) for \c - 1\ &lt;&lt; 1. Both in d = 1 and 2, we obtain theta = 1.

    DOI: 10.1088/0305-4470/32/9/011

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  • Chiral Potts models, friendly walkers and directed percolation problem Reviewed

    T Tsuchiya, M Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   67 ( 5 )   1655 - 1666   1998.5

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    The X-state chiral Potts model on a finite directed lattice is defined: whose partition function under a certain boundary condition becomes the directed percolation (DP) probability for a finite lattice in the lambda --&gt; 1 limit. We also introduce the system of m friendly walkers of L time-steps and prove that its generating function of trajectories is equal to the partition function of the X-state chiral Potts model when m = (lambda-1)/2. Combining these results gives a new formula for the DP probability given by a double limit L --&gt; infinity and m --&gt; O of the generating function of the m friendly walkers. We define the critical value p(c)((m)) for the infinite system of. m friendly walkers. Numerical study supports our conjecture that the critical value p(c), for the DP probability is given by the m --&gt; O extrapolation of p(c)((m)).

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  • Forest dynamics with canopy gap expansion and stochastic Ising model Reviewed

    M Katori, S Kizaki, Y Terui, T Kubo

    FRACTALS-AN INTERDISCIPLINARY JOURNAL ON THE COMPLEX GEOMETRY OF NATURE   6 ( 1 )   81 - 86   1998.3

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    Importance of the influence of neighboring canopy gaps upon new gap creation has been clarified by the ecological study of a neotropical forest on Barro Colorado Island (BCI), Panama. A stochastic lattice model for the forest dynamics with interacting canopy gap expansion was introduced by Kubo et al. We give a theorem showing a condition that this model can be regarded as a stochastic Ising model, and that its stationary state is exactly given by a Gibbs state. Using this theorem, we obtain a Gibbs state which remarkably well approximates the real gap-size distribution in BCI.

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  • n-state exclusive diffusion models for avalanche processes showing self-organized criticality Reviewed

    H Kobayashi, M Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   66 ( 8 )   2367 - 2382   1997.8

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    We introduce new avalanche models on the d-dimensional hyper-cubic lattices which show the self-organized criticality (SOC). They are generalization of the two-state exclusive diffusion model introduced in our previous paper and called the n-state exclusive diffusion models (1 less than or equal to n less than or equal to 2d). If n = 2d, this model is the sandpile model of Bak, Tang and Wiesenfeld. By Monte Carlo simulations, we evaluated the height distributions of particles and the avalanche exponents for d = 2, 3, and 4. Numerical results imply that the n-state exclusive diffusion models belong to the same universality class for each d, if 2 less than or equal to n less than or equal to 2d. We apply the mean-field approximation and evaluate the height distributions of particles in the SOC states. It is shown that the mean-field theory by Tang and Bak is the d --&gt; infinity limit of our mean-field approximation.

    DOI: 10.1143/JPSJ.66.2367

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  • The number of directed compact site animals and extrapolation formula of directed percolation probability Reviewed

    N Inui, M Katori, G Komatsu, K Kameoka

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   66 ( 5 )   1306 - 1309   1997.5

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    We consider a quantity, alpha(n,m) = the number of directed compact site animals with m sites at the height n. We give an explicit expression for alpha(n,m) and calculate the mean width and the variance of width at the height n of directed compact animals exactly. The number of alpha(n,m) is identified with a coefficient appearing in the series expansion for the directed site percolation. An extrapolation formula of directed site percolation probability for the square lattice is obtained using this number. The generating function of alpha(n,m) is calculated and some summation formulae are given.

    DOI: 10.1143/JPSJ.66.1306

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  • Ballot number representation of the percolation probability series for the directed square lattice Reviewed

    M Katori, N Inui

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   30 ( 9 )   2975 - 2994   1997.5

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    Series expansion data are matters of increasing importance far studying the directed percolation problem and others which are not yet solved. In order to extrapolate series for the percolation probability on the directed square lattice, Baxter and Guttmann proposed a numerical method based on an assumption that the so-called correction terms are expressed as rational functions of the Catalan numbers. We give a theorem that the coefficients of the series are generally given as finite series of the ballot numbers, which proves the assumption by Baxter and Guttmann as a corollary. The proof of the theorem gives a method to calculate correction terms exactly, as demonstrated by calculating the first three correction terms explicitly. Although the present work provides a mathematical basis for the extrapolation procedure, there are still open problems concerning this procedure.

    DOI: 10.1088/0305-4470/30/9/012

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  • Anomalous heat capacity of antiferromagnet FeBr2 in a magnetic field Reviewed

    HA Katori, K Katsumata, M Katori

    JOURNAL OF APPLIED PHYSICS   81 ( 8 )   4396 - 4398   1997.4

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    Heat capacity measurements in external magnetic fields (H) have been performed on the Ising antiferromagnet FeBr2. The temperature dependence of the magnetic heat capacity shows two anomalies for 1.4 T less than or equal to H less than or equal to 2.9 T. The anomaly at a higher temperature indicates the phase transition from the paramagnetic to antiferromagnetic phase. The anomaly at a lower temperature is formed by a sharp peak superposed on a broad shoulder. The existence of two peaks shows the occurrence of a new phase transition in addition to the antiferromagnetic transition in magnetic fields. A Monte Carlo simulation for the heat capacity reproduces the peak indicating the antiferromagnetic transition and the bread shoulder. The nonuniform spin configurations in the layers with negative moment, which is caused by the competition between the nearest-neighbor ferromagnetic and the next-nearest-neighbor antiferromagnetic interactions in the triangular Fe-planes, is shown to produce the broad shoulder. (C) 1997 American Institute of Physics.

    DOI: 10.1063/1.364836

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  • Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice Reviewed

    M Katori, N Inui, G Komatsu, K Kameoka

    JOURNAL OF STATISTICAL PHYSICS   86 ( 1-2 )   37 - 55   1997.1

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    The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameter k is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term d(n,l)(k) is expressed by Gauss' hypergeometric series with a variable k. Since the ADBP includes the ordinary directed bond percolation as a special case with k = 1, our results give another proof for the Baxter-Guttmann's conjecture that d(n,1)(1) is given by the Catalan number, which was recently proved by Bousquet-Melou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis by Dlog Pade approximations suggests that the critical value depends on k, while asymmetry does not change the critical exponent beta of percolation probability.

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  • Specific-heat anomaly in the Ising antiferromagnet FeBr2 in external magnetic fields Reviewed

    HA Katori, K Katsumata, M Katori

    PHYSICAL REVIEW B   54 ( 14 )   R9620 - R9623   1996.10

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    Specific-heat measurements have been performed on the Ising antiferromagnet FeBr2 in external magnetic fields. The temperature dependence of the magnetic specific heat shows two anomalies in external fields below the metamagnetic transition field. The anomaly at a higher temperature indicates the phase transition from the paramagnetic to antiferromagnetic phase. The lower-temperature anomaly shows up in the form of a peak superposed on a broad shoulder. The peak becomes sharp with the increase of magnetic field up to 2.9 T, which shows the existence of a new phase under a magnetic field. A theoretical analysis based on the pair approximation and Monte Carlo simulations reproduces the peak at the higher temperature and the broad shoulder. The broad shoulder appears as a result of competition between the nearest-neighbor ferromagnetic and the next-nearest-neighbor antiferromagnetic interactions in the triangular Fe lattice planes.

    DOI: 10.1103/PhysRevB.54.R9620

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  • Time-reversal duality and planar lattice duality in non-equilibrium lattice models Reviewed

    M Katori

    FRACTALS-AN INTERDISCIPLINARY JOURNAL ON THE COMPLEX GEOMETRY OF NATURE   4 ( 3 )   285 - 292   1996.9

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    The contact process (CP) is a simple mathematical model for the spread of infection of a contagious disease. Though it has only nearest-neighbor interactions, phase transitions occur even in the one-dimensional system and non-equilibrium stationary states appear in supercritical phase. This implies violation of detailed balance. The appearance of such non-equilibrium states is related to directed percolation problems on the spatio-temporal plane. In the present paper, we study discrete-time versions of the CP, the two-neighbor stochastic cellular automata (SCA), and clarify this viewpoint. We use two kinds of duality relations, the time-reversal duality and the planar lattice duality on the spatio-temporal plane, and give a good lower bound for the critical line of non-equilibrium phase transitions in the two-neighbor SCA.

    DOI: 10.1142/S0218348X9600039X

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  • Exclusive diffusion model showing self-organized criticality Reviewed

    H Kobayashi, M Katori

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   65 ( 8 )   2536 - 2542   1996.8

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    We introduce a new model which shows the self-organized criticality (SOC). We call it the exclusive diffusion model, since if two or more than two particles are put on a site, they exclusively diffuse to different nearest-neighbor sites. It can be regarded as a modified process of the two-state model introduced by Manna. By numerical simulations we evaluated the particle densities and various critical exponents in the SOC states for the square lattice and the simple cubic lattice. The particle density of the exclusive diffusion model is different from the value reported by Manna for the two-state model in two dimensions, while it is concluded that this model and the two-state model belong to the same universality class. We show that the mean-field approximation, which we proposed for the sandpile models in a previous paper, is also applicable to the exclusive diffusion model.

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  • Mean-field theory of avalanches in self-organized critical states Reviewed

    M Katori, H Kobayashi

    PHYSICA A   229 ( 3-4 )   461 - 477   1996.8

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    A new mean-field approximation is proposed for the sandpile model of Bak, Tang and Wiesenfeld and the distribution of heights in the self-organized critical state is calculated. We treat several series of successive topples to approximate various avalanches. In contrast to the previous mean-field theory given by Tang and Bak, which assumes the stationary condition among topples in an avalanche, our theory takes into account long-term behavior of the system. The results are in good agreement with the values estimated by computer simulations even in low dimensions d = 2 and 3. Higher approximations obtained by including local correlations or by imposing a reducibility condition, which is used in the cluster variation method, are also shown.

    DOI: 10.1016/0378-4371(96)00003-9

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  • Catalan numbers in a series expansion of the directed percolation probability on a square lattice Reviewed

    N Inui, M Katori

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   29 ( 15 )   4347 - 4364   1996.8

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    We regard the bond directed percolation on a square lattice as a discrete-time Markov process of a one-dimensional interacting particle system. The coefficients in series expansion of the probability P-n,P-m of having m particles at time n - 1 are studied. We derive the difference equations for the first and the second series of coefficients and prove that these coefficients are expressed using the ballot numbers, whose special cases are known as the Catalan numbers. As a corollary of our results, we prove a part of the conjecture by Baxter and Guttmann that the correction terms are expressed as rational functions of the Catalan numbers. We also give approximations for the percolation probability using the present formula.

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  • Specific-Heat Anomaly in the Ising Antiferromagnet FeBr2 in External Magnetic Fields

    KATORI H A, KATSUMATA K, KATORI M

    Physical Review B   54 ( 14 )   R9620 - R9623   1996

  • Specific-heat anomaly in the Ising antiferromagnet F in external magnetic fields Reviewed

    H. Aruga Katori, K. Katsumata, M. Katori

    Physical Review B - Condensed Matter and Materials Physics   54 ( 14 )   R9620 - R9623   1996

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    Specific-heat measurements have been performed on the Ising antiferromagnet Fe(Formula presented) in external magnetic fields. The temperature dependence of the magnetic specific heat shows two anomalies in external fields below the metamagnetic transition field. The anomaly at a higher temperature indicates the phase transition from the paramagnetic to antiferromagnetic phase. The lower-temperature anomaly shows up in the form of a peak superposed on a broad shoulder. The peak becomes sharp with the increase of magnetic field up to 2.9 T, which shows the existence of a new phase under a magnetic field. A theoretical analysis based on the pair approximation and Monte Carlo simulations reproduces the peak at the higher temperature and the broad shoulder. The broad shoulder appears as a result of competition between the nearest-neighbor ferromagnetic and the next-nearest-neighbor antiferromagnetic interactions in the triangular Fe lattice planes. © 1996 The American Physical Society.

    DOI: 10.1103/PhysRevB.54.R9620

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  • 2-NEIGHBOR STOCHASTIC CELLULAR-AUTOMATA AND THEIR PLANAR LATTICE DUALS Reviewed

    M KATORI, H TSUKAHARA

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   28 ( 14 )   3935 - 3957   1995.7

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    Two-neighbour stochastic cellular automata (SCA) are the set of one-dimensional discrete-time interacting particle systems with two parameters, which show non-equilibrium phase transitions from the extinction phase to the survival phase. The phase diagram was first studied by Kinzel using a numerical method called transfer-matrix scaling. For some parameter region the processes can be defined as directed percolation models on the spatio-temporal plane and the bond- and site-directed percolation models are included as special cases. Extending the argument of Dhar, Barma and Phani originally given for bond-directed percolation, we introduce diode-resistor percolation models which are the planar lattice duals of the SCA and give rigorous lower bounds for the critical line. In special cases, our results give 0.6885 less than or equal to alpha(c) and 0.6261 less than or equal to beta(c), where alpha(c) and beta(c) denote the critical probabilities of the site- and bond-directed percolation models on the square lattice. respectively. Combining the upper bound for the critical line recently proved by Liggett, we summarize the rigorous results for the phase diagram of the systems. Results of computer simulation are also shown.

    DOI: 10.1088/0305-4470/28/14/014

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  • DUALITY AND UNIVERSALITY IN NONEQUILIBRIUM LATTICE MODELS Reviewed

    N INUI, M KATORI, T UZAWA

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   28 ( 7 )   1817 - 1830   1995.4

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    We introduce two kinds of discrete-time non-equilibrium lattice models (stochastic cellular automata), which we call the creation process eta(t) and the branching process &lt;(eta)over bar&gt;(t). In special cases the former process can be identified with the site or bond directed percolation models. When the system is defined on a d-dimensional finite lattice with size L, these processes are determined by 2(Ld) x 2(Ld) transition matrices, M(L) and M(L), respectively. It is proved that subject to certain relations between the parameters of these models, M(L) and the transpose of M(L) are conjugate and thus the characteristic polynomials become equal to each other, det(M(L) - lambda E) = det(M(L) - lambda E), for arbitrary L greater than or equal to n, where E is the identity matrix. Since dynamical critical exponents as well as critical values will be determined by the asymptotic behaviour in the limit L --&gt; infinity of the large eigenvalues of the transition matrix, our result implies that, if continuous phase transitions and critical phenomena are observed, these two processes belong to the same universality class. In proving the equality, we use the relation which is called the coalescing duality in probability theory.

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  • RIGOROUS RESULTS FOR THE DIFFUSIVE CONTACT-PROCESSES IN D-GREATER-THAN-OR-EQUAL-TO-3 Reviewed

    M KATORI

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   27 ( 22 )   7327 - 7341   1994.11

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    The diffusive contact process is an interacting particle system on the d-dimensional hypercubic lattice. Each site can be occupied by, at most, one particle and each particle can do the following three things. (i) At rate 1 a particle will be annihilated. (ii) At rate lambda a particle will give birth to a new particle at one of its 2d neighbour sites, if it is vacated. (iii) At rate D a particle will hop to one of its 2d neighbour sites, if it is vacated. For each D greater-than-or-equal-to 0, there is a critical value lambda(c)(d) (D) so that for lambda(c)(d) (D) all particles will be annihilated with probability 1 while for lambda &gt; lambda(c)(d) (D) particles will survive with a positive probability even in the limit t --&gt; infinity. In the present paper the lower and upper bounds for lambda(c)(d) (D) are given (theorems 1.1 and 1.2) and a lower bound for the density of particles is given in the case lambda &gt; lambda(c)(d) (D) (theorem 2.2), when the dimensionality d greater-than-or-equal-to 3. Rigorous results concluded from these theorems are shown, The crossover phenomenon for large D is discussed for the three-dimensional case.

    DOI: 10.1088/0305-4470/27/22/010

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  • STRUCTURAL AND STATISTICAL PROPERTIES OF COMPETING DIRECTED PERCOLATION Reviewed

    S ISOGAMI, M KATORI, M MATSUSHITA

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   63 ( 8 )   2919 - 2929   1994.8

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    We studied competing directed percolation (CDP) to confirm the validity of Matsushita and Meakin's theory also for the case that an entire pattern is self-similar and not compact, while its individual clusters are self-affine. Critical values of the bond concentration p(c) and values of the various scaling exponents of CDP in 2 approximately 6 dimensions were estimated by simulations. The present result shows that Matsushita and Meakin's theory is valid for CDP in any dimension. It is also found that the estimated values of the scaling exponent of the cluster-size distribution tau are different from those of ordinary directed percolation.

    DOI: 10.1143/JPSJ.63.2919

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  • REFORMULATION OF GRAYS DUALITY FOR ATTRACTIVE SPIN SYSTEMS AND ITS APPLICATIONS Reviewed

    M KATORI

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   27 ( 9 )   3191 - 3211   1994.5

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    Duality relations, which associate two Markov processes in different state spaces, have been useful tools in the study of the long-term behaviour of the stochastic processes of spin systems. In 1986, Gray introduced a new duality theory which is applicable to general spin systems with attractive transition rates. The theory was developed by making full use of graphical representations. In the present paper, Gray's duality is reformulated by studying the action of the formal generators on a newly chosen duality function and the dual processes defined in the state space UPSILON = {A : A is a finite subset of Y) are discussed. where Y is a collection of finite subsets of Z(d). As applications of the submodularity of the survival probability sigma(A) of the dual processes, rigorous lower bounds of the critical values are derived for the theta-contact process, the multi-particle creation model and the sexual reproduction process.

    DOI: 10.1088/0305-4470/27/9/030

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  • ON THE CAM CANNONICALITY OF THE CLUSTER-VARIATION APPROXIMATIONS Invited Reviewed

    M KATORI, M SUZUKI

    PROGRESS OF THEORETICAL PHYSICS SUPPLEMENT   115 ( 115 )   83 - 93   1994

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    The cluster-variation method (CVM) proposed by Kikuchi is a general theory to give approximations, which are useful to discuss phase transitions qualitatively in many systems. Recently one of the present authors (M. S.) proposed the coherent-anomaly method (CAM). If we have a well-behaved series of approximations, which is called a canonical series, we can estimate critical exponents following the CAM. In this paper it is demonstrated by using the ferromagnetic Ising models that the CVM will provide a canonical series which shows coherent anomaly. Our result implies that combining the CVM with the CAM will give a powerful method to study phase transitions and critical phenomena.

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  • BOUNDS FOR THE CRITICAL LINE OF THE THETA-CONTACT PROCESSES WITH 1-LESS-THAN-OR-EQUAL-TO-THETA-LESS-THAN-OR-EQUAL-TO-2 Reviewed

    M KATORI, N KONNO

    JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL   26 ( 23 )   6597 - 6614   1993.12

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    We study a family of the one-dimensional contact processes introduced by Durrett and Griffeath, which is parametrized by theta. For each theta greater-than-or-equal-to 1, there is a unique critical value lambda(c) (theta) so that any process becomes extinct with probability 1 for lambda &lt; lambda(c) (theta), but all processes starting from non-empty initial states have positive probabilities of survival for lambda &gt; lamba(c)(theta). In this paper we give rigorous upper and lower bounds for the critical line lambda = lambda(c)(theta) for 1 less-than-or-equal-to theta less-than-or-equal-to 2. In order to obtain the upper bound we extend the Holley-Liggett argument which was originally given for the case theta = 2 (the basic contact process). We construct a new class of attractive renewal measures with positive densities and the upper bound of lambda(c)(theta) is given as the largest root of a cubic equation, thetalambda3 - (3theta - 2)lambda2 - 3(2 - theta)lambda + (2 - theta) = 0. Recently Liggett reported an upper bound of the critical value for the case theta = 1 (the threshold contact process) by a modified version of the Holley-Liggett argument. Our result includes these previous results as special cases.

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  • ON THE EXTINCTION OF DICKMAN REACTION - DIFFUSION-PROCESSES Reviewed

    M KATORI, N KONNO

    PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS   186 ( 3-4 )   578 - 590   1992.8

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    Some comments on the extinction of processes are given for the reaction-diffusion processes recently proposed by Dickman as mathematical models for chemical reactions on catalytic surfaces. A brief review of mean-field-type approximations (MFA) is presented for three models; the single annihilation model (SAM), the pair annihilation model (PAM) and the triplet annihilation model (TAM). Two theorems on the extinction are proved. The former supports the MFA predictions for the SAM. The latter gives a qualitative correction to the phase diagram obtained by the MFA for the PAM in low dimensions (d less-than-or-equal-to 2). In order to obtain the latter theorem, we discuss the relationship between the PAM and the branching annihilating random walk of Bramson and Gray.

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  • UPPER-BOUNDS FOR ORDER PARAMETERS OF A CLASS OF ATTRACTIVE NEAREST-PARTICLE SYSTEMS WITH FINITE RANGES Reviewed

    N KONNO, M KATORI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   61 ( 3 )   806 - 811   1992.3

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    A class of the attractive nearest-particle systems with finite ranges is studied. By using the correlation identities and the FKG inequality, the rigorous upper bounds rho(lambda)(2) for the order parameters of this class are obtained. Furthermore, it is shown that the rigorous lower bounds lambda(c)(2) for the critical values of it are strictly greater than one.

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  • PHASE-TRANSITIONS IN SPIN SYSTEMS WITHOUT DETAILED BALANCE

    M KATORI, N KONNO

    JOURNAL OF MAGNETISM AND MAGNETIC MATERIALS   104   267 - 268   1992.2

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    Stochastic Ising models for phase transitions in magnetic systems satisfy the condition of detailed balance and thus they have the Gibbs states as stationary states. On the other hand, some spin systems which do not satisfy the condition of detailed balance can have nontrivial stationary states as a result of nonequilibrium phase transitions. In the present paper, the contact process of Harris and the reaction-diffusion processes of Dickman are studied as typical examples of this class. The survival probability sigma(A) is introduced to study the stationary states. Some theorems are given for these processes.

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  • ANALYSIS OF THE ORDER PARAMETER FOR UNIFORM NEAREST PARTICLE SYSTEM Reviewed

    N KONNO, M KATORI

    JOURNAL OF STATISTICAL PHYSICS   65 ( 1-2 )   247 - 254   1991.10

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    The uniform nearest particle system (UNPS) is studied, which is a continuous-time Markov process with state space {0,1}Z1. The rigorous upper bound rho-lambda(mf) = (lambda-1)/lambda for the order parameter rho-lambda is given by the correlation identity and the FKG inequality. Then an improvement of this bound rho-lambda(mf) is shown in a similar fashion; rho-lambda less-than-or-equal-to C(lambda-1)/log(lambda-1)\for lambda &gt; 1. Recently, Mountford proved that the critical value lambda-c = 1. Combining his result and our improved bound implies that if the critical exponent-beta exists, it is strictly greater than the mean-field value 1 in the weak sense.

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  • MULTI-EFFECTIVE-FIELD THEORY - APPLICATIONS TO THE CAM ANALYSIS OF THE 2-DIMENSIONAL ISING-MODEL Reviewed

    K MINAMI, Y NONOMURA, M KATORI, M SUZUKI

    PHYSICA A   174 ( 2-3 )   479 - 503   1991.6

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    A multi-effective-field theory is formulated and applied to the two-dimensional Ising model from the viewpoint of the coherent-anomaly method (CAM). Two necessary conditions to construct the CAM canonical series are shown. Two series of approximations are derived on 3 x 3- and 4 x 4-clusters and with the use of the CAM we estimate the critical exponent of the susceptibility within an error of 0.37 and 0.45 percent, respectively, for the exact value gamma = 1.75 where the exact T(c)* is assumed. This accuracy of the estimation shows the effectiveness of this theory. It is generally proved that certain kinds of approximations with different combinations of effective fields yield exactly the same approximate critical temperature and mean-field coefficient.

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  • UPPER-BOUNDS FOR SURVIVAL PROBABILITY OF THE CONTACT PROCESS Reviewed

    M KATORI, N KONNO

    JOURNAL OF STATISTICAL PHYSICS   63 ( 1-2 )   115 - 130   1991.4

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    A precise description of the nontrivial upper invariant measure for lambda &gt; lambda-c is still an open problem for the basic contact process, which is a self-dual, attractive, but nonreversible Markov process of an interacting particle system. By its self-duality, to identify the invariant measure is equivalent to determining the initial-state dependence of the survival probability of the process. A procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by Harris. Two new bounds are given, improving the simple branching-process bound. In the one-dimensional case, the present procedure can be viewed as a trial to make approximate measures by generalized Markov extensions.

    DOI: 10.1007/BF01026595

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  • 3-POINT MARKOV EXTENSION AND AN IMPROVED UPPER BOUND FOR SURVIVAL PROBABILITY OF THE ONE-DIMENSIONAL CONTACT PROCESS Reviewed

    M KATORI, N KONNO

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   60 ( 2 )   418 - 429   1991.2

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    The survival probability sigma(A) is studied for the one-dimensional contact process. An approximate function h(3) (A) for sigma(A) is obtained; a partial stationary condition is used to determine h(3) (C(i)) for small sets C(i)'s and then it is extended to all finite subsets A of Z1 by the Markov extension. It is proved that h(3)(A) gives an upper bound for sigma(A) in the parameter region where the infection rate lambda is greater than Griffeath's lower bound for lambda-c, i.e. lambda &gt; 1/6 (1 + square-root 37). The present result improves previous upper bounds.

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  • APPLICATIONS OF THE HARRIS-FKG INEQUALITY TO UPPER-BOUNDS FOR ORDER PARAMETERS IN THE CONTACT-PROCESSES Reviewed

    N KONNO, M KATORI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   60 ( 2 )   430 - 434   1991.2

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    Upper bounds for order parameters are studied for the d-dimensional contact processes by using the Harris-FKG inequality. Particularly, in the one-dimensional system, it is proved that our improved upper bound rho-1(4), lambda is less than Griffeath's bound for lambda greater-than-or-equal-to lambda-c(1).

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  • Correlation Identities for Nearest-Particle Systems and their Applications to One-Dimentional Contact Proces Reviewed

    N.Konno, M.Katori

    Modern Physics Letters   B5 ( 2 )   151 - 159   1991.2

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    DOI: 10.1142/S0217984991000198

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  • AN UPPER BOUND FOR SURVIVAL PROBABILITY OF INFECTED REGION IN THE CONTACT-PROCESSES Reviewed

    M KATORI, N KONNO

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   60 ( 1 )   95 - 99   1991.1

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    An upper bound for the survival probability of infected region sigma-d,lambda (A), (A is-contained-in Z(d)), is presented for the contact process. The bound depends not only on the cardinality number \A\ but also on the number of neighboring pairs of sites in A. It may correspond to Krinsky's bound using the Bethe approximation for the Ising models. The present result can be viewed as a generalization of Griffeath's bound for an order parameter pl,lambda (= sigma-1-lambda({x})) in the one-dimensional system to the survival probabilities for all A in arbitrary dimensions.

    DOI: 10.1143/JPSJ.60.95

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  • STUDY OF COHERENT ANOMALIES AND CRITICAL EXPONENTS BASED ON HIGH-LEVEL CLUSTER-VARIATION APPROXIMATIONS Reviewed

    S FUJIKI, M KATORI, M SUZUKI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   59 ( 8 )   2681 - 2687   1990.8

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    DOI: 10.1143/JPSJ.59.2681

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  • Systematic Approach to Critical Phenomena by the Extended Variational Method and Coherent-Anomaly Method Reviewed

    N.Kawashima, M.Katori, C.Tsallis, M.Suzuki

    International Journal of Modern Physics   B4 ( 7&8 )   1409 - 1422   1990.7

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  • APPLICATIONS OF THE CAM BASED ON A NEW DECOUPLING PROCEDURE OF CORRELATION-FUNCTIONS IN THE ONE-DIMENSIONAL CONTACT PROCESS Reviewed

    N KONNO, M KATORI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   59 ( 5 )   1581 - 1592   1990.5

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    DOI: 10.1143/JPSJ.59.1581

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  • CORRELATION INEQUALITIES AND LOWER BOUNDS FOR THE CRITICAL VALUE LAMBDA-C OF CONTACT-PROCESSES Reviewed

    M KATORI, N KONNO

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   59 ( 3 )   877 - 887   1990.3

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    DOI: 10.1143/JPSJ.59.877

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  • A 2ND-ORDER PHASE-TRANSITION AS A LIMIT OF THE 1ST-ORDER PHASE-TRANSITIONS - COHERENT ANOMALIES AND CRITICAL PHENOMENA IN THE POTTS MODELS Reviewed

    M KATORI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   57 ( 12 )   4114 - 4125   1988.12

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    DOI: 10.1143/JPSJ.57.4114

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  • STUDY OF COHERENT ANOMALIES AND CRITICAL EXPONENTS BASED ON THE CLUSTER-VARIATION METHOD Reviewed

    M KATORI, M SUZUKI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   57 ( 11 )   3753 - 3761   1988.11

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    DOI: 10.1143/JPSJ.57.3753

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  • Coherent-Anomaly Method in Critical Phenomena Reviewed

    M.Katori, M.Suzuki

    Progress in Statistical Mechanics ed. By C. -K.Hu   273 - 287   1988.10

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  • COHERENT-ANOMALY METHOD IN CRITICAL PHENOMENA .5. ESTIMATION OF THE DYNAMICAL CRITICAL EXPONENT-DELTA OF THE TWO-DIMENSIONAL KINETIC ISING-MODEL Reviewed

    M KATORI, M SUZUKI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   57 ( 3 )   807 - 817   1988.3

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    DOI: 10.1143/JPSJ.57.807

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  • COHERENT-ANOMALY METHOD IN CRITICAL PHENOMENA .3. MEAN-FIELD TRANSFER-MATRIX METHOD IN THE 2D ISING-MODEL Reviewed

    HU, X, M KATORI, M SUZUKI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   56 ( 11 )   3865 - 3880   1987.11

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    DOI: 10.1143/JPSJ.56.3865

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  • COHERENT ANOMALY METHOD IN CRITICAL PHENOMENA .1.

    M SUZUKI, M KATORI, HU, X

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   56 ( 9 )   3092 - 3112   1987.9

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    DOI: 10.1143/JPSJ.56.3092

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  • COHERENT ANOMALY METHOD IN CRITICAL PHENOMENA .2. APPLICATIONS TO THE TWO-DIMENSIONAL AND 3-DIMENSIONAL ISING-MODELS Reviewed

    M KATORI, M SUZUKI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   56 ( 9 )   3113 - 3125   1987.9

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    DOI: 10.1143/JPSJ.56.3113

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  • NEW METHOD TO STUDY CRITICAL PHENOMENA - MEAN-FIELD FINITE-SIZE SCALING THEORY Reviewed

    M SUZUKI, M KATORI

    JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN   55 ( 1 )   1 - 4   1986.1

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    DOI: 10.1143/JPSJ.55.1

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  • SCALING LAWS OF THE NONLINEAR PART OF MAGNETIZATION IN THE SK MODEL OF SPIN-GLASSES Reviewed

    M KATORI, M SUZUKI

    PROGRESS OF THEORETICAL PHYSICS   74 ( 6 )   1175 - 1190   1985.12

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    DOI: 10.1143/PTP.74.1175

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Books

  • Elliptic Extensions in Statistical and Stochastic Systems Reviewed

    Makoto Katori( Role: Sole authorall pages (pp.125))

    Springer Singapore2  2023.4  ( ISBN:9789811995279

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    Total pages:125   Responsible for pages:125   Language:English   Book type:Scholarly book

    Hermite's theorem makes it known that there are three levels of mathematical frames in which a simple addition formula is valid. They are rational, q-analogue, and elliptic-analogue. Based on the addition formula and associated mathematical structures, productive studies have been carried out in the process of q-extension of the rational (classical) formulas in enumerative combinatorics, theory of special functions, representation theory, study of integrable systems, and so on. Originating from the paper by Date, Jimbo, Kuniba, Miwa, and Okado on the exactly solvable statistical mechanics models using the theta function identities (1987), the formulas obtained at the q-level are now extended to the elliptic level in many research fields in mathematics and theoretical physics. In the present monograph, the recent progress of the elliptic extensions in the study of statistical and stochastic models in equilibrium and nonequilibrium statistical mechanics and probability theory is shown. At the elliptic level, many special functions are used, including Jacobi's theta functions, Weierstrass elliptic functions, Jacobi's elliptic functions, and others. This monograph is not intended to be a handbook of mathematical formulas of these elliptic functions, however. Thus, use is made only of the theta function of a complex-valued argument and a real-valued nome, which is a simplified version of the four kinds of Jacobi's theta functions. Then, the seven systems of orthogonal theta functions, written using a polynomial of the argument multiplied by a single theta function, or pairs of such functions, can be defined. They were introduced by Rosengren and Schlosser (2006), in association with the seven irreducible reduced affine root systems. Using Rosengren and Schlosser's theta functions, non-colliding Brownian bridges on a one-dimensional torus and an interval are discussed, along with determinantal point processes on a two-dimensional torus. Their scaling limits are argued, and the infinite particle systems are derived. Such limit transitions will be regarded as the mathematical realizations of the thermodynamic or hydrodynamic limits that are central subjects of statistical mechanics.

    DOI: 10.1007/978-981-19-9527-9

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  • 例題から展開する熱・統計力学

    香取, 眞理, 森山, 修( Role: Joint author全文共同執筆)

    サイエンス社  2021.10  ( ISBN:9784781915234

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    Total pages:v, 215p   Language:Japanese   Book type:Textbook, survey, introduction

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  • 例題から展開する電磁気学

    サイエンス社  2018.7 

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  • 例題から展開する力学

    サイエンス社  2017.5 

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  • Progress in Nanophotonics 4, Chapter 2 Nonequilibrium Statistical Mechanical Models for Photon Breeding Processes Assisted by Dressed-Photon-Phonons

    Springer  2017.1 

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  • Progress in Nanophotonics 4, Chapter 2 Nonequilibrium Statistical Mechanical Models for Photon Breeding Processes Assisted by Dressed-Photon-Phonons

    Springer  2017.1 

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  • 詳解と演習 大学院入試問題 「物理学」

    数理工学社  2016.3 

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  • Bessel Processes, Schramm-Loewner Evolution, and the Dyson Model

    Makoto Katori( Role: Sole author)

    Springer  2015.12 

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  • 統計力学

    裳華房  2010.11 

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  • 問題例で深める物理---統一的思考力を身につけるために---

    サイエンス社  2010.5 

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  • 統計物理学ハンドブック ---熱平衡から非平衡まで---

    朝倉書店  2007.6 

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  • ランダム行列と非衝突過程

    遊星社  2006.8 

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  • 科学技術者のための数学ハンドブック

    朝倉書店  2002.9 

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  • 物理数学の基礎

    サイエンス社  2001.4 

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  • 非平衡統計力学

    裳華房  1999.3 

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  • 複雑系を解く確率モデル(217頁)

    講談社  1997.11 

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  • Coherent-Anomaly Method: Mean-Field, Fluctuations and Systematics(515pages)(Chapter3,Chapter8を執筆)

    World Scientific  1995.7 

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  • Coherent-Anomaly Method: Mean-Field, Fluctuations and Systematics(515pages)(Chapter3,Chapter8)

    World Scientific, Singapore  1995.7 

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MISC

  • ランダム行列とはなにか Invited

    香取 眞理

    数学セミナー   58 ( 2(688) )   8 - 12   2019.2

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  • Complex systems of dressed photons and applications

    Motoichi Ohtsu, Makoto Katori

    The Review of Laser Engineering   139 - 143   2017.3

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  • Complex System of Dressed Photons and Applications Reviewed

    45 ( 3 )   139 - 143   2017.3

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  • 新著紹介:小野嘉之著「初歩の統計力学を取り入れた熱力学」

    香取眞理

    日本物理学会誌   72 ( 2 )   p.140   2017.2

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  • Characterization of the hydrodynamic limit of the Dyson model Reviewed

    Sergio Andraus, Makoto Katori

    RIMS Kokyuroku Bessatsu   B59 ( 1 )   157 - 173   2016.7

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  • 偶然を科学する(統計力学と確率論) Invited

    香取眞理

    数理科学   54 (No.634) ( 4 )   30 - 36   2016.4

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  • Abelian Sandpile Models in Statistical Mechanics -- Dissipative Abelian Sandpile Models --

    Makoto Katori

    MI Lecture Note Series   63   58 - 91   2015.8

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  • 10aAR-4 Collective motion of bacterial cells in a two-dimensional circular pool

    Tsukamoto Shota, Yamamoto Ken, Katori Makoto, Wakita Jun-ichi

    Meeting abstracts of the Physical Society of Japan   69 ( 2 )   226 - 226   2014.8

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  • Height correlation of ripped graphene and Lundberg-Folk formula for magnetoresistance Reviewed

    Kazuyuki Gemma, Makoto Katori

    Journal of Institute of Science and Engineering, Chuo University   19 ( 19 )   1 - 21   2014.3

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    Other Link: http://ir.c.chuo-u.ac.jp/repository/search/item/md/rsc/p/8501/

  • 統計力学と関数論とブラウン運動と

    香取眞理

    数理科学   600 ( 6 )   44 - 49   2013.6

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  • 26pXJ-3 Height correlation of rippled graphene and Lundeberg-Folk formula for magnetoresistance

    Genma Kazuyuki, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   68 ( 1 )   752 - 752   2013.3

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  • 26aXR-6 Monte Carlo simulations of Brownian motion models with Toda-lattice interactions

    Tada Hiroki, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   68 ( 1 )   321 - 321   2013.3

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  • 26pXR-9 Stieltjes-Wigert ensemble and q-Painleve II equation

    Takahashi Yuta, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   68 ( 1 )   335 - 335   2013.3

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  • 26aXR-8 TASEP on a circle and observations of circular motions of bacteria

    Yamada Yasuyuki, Katori Makoto, Wakita Jun-ichi

    Meeting abstracts of the Physical Society of Japan   68 ( 1 )   322 - 322   2013.3

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  • 19pAC-5 Noncolliding Brownian Motion with Drift and Stieltjes-Wigert Ensemble

    Takahashi Yuta, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   67 ( 2 )   269 - 269   2012.8

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  • 26pAC-5 Percolation analysis of the Cosmic Microwave Background radiation

    Higuchi M., Kobayashi N., Wakita J., Yamazaki Y., Kuninaka H., Katori M., Matsushita M., Matsushita S., Chiang Lung-Yih

    Meeting abstracts of the Physical Society of Japan   67 ( 1 )   344 - 344   2012.3

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  • 26pAE-15 Intertwining Operator of Dunkl Processes and Dyson's Model

    Andraus S., Katori M., Miyashita S.

    Meeting abstracts of the Physical Society of Japan   67 ( 1 )   349 - 349   2012.3

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  • 26pAE-13 Power matrix approach for Schramm Loewner Evolution

    Kawahara Yomei, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   67 ( 1 )   349 - 349   2012.3

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  • 26pAE-14 Fractal dimensions and winding angle distributions of random curves

    Nishikiori Yukina, Kobayasi Naoki, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   67 ( 1 )   349 - 349   2012.3

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  • The Calogero-Moser System with Particle Exchange and Dunkl Processes

    ANDRAUS S., 香取眞理, 宮下精二

    日本物理学会講演概要集   67 ( 2 )   2012

  • 23aGT-12 Cosmic Microwave Background Radiation and Loop Statistics

    Kobayashi Naoki, Yamazaki Yoshihiro, Kuninaka Hiroto, Katori Makoto, Matsushita Mitsugu, Matsushita Satoki, Chiang Lung-Yih

    Meeting abstracts of the Physical Society of Japan   66 ( 2 )   273 - 273   2011.8

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  • 22aGE-12 The velocity correlation function of the totally asymmetric simple exclusion process on a ring and the Gauss hypergeometric function

    Yamada Yasuyuki, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   66 ( 2 )   221 - 221   2011.8

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  • Iterative Schwarz-Christoffel transformations driven by random walks and fractal curves Reviewed

    Fumihito Sato, Makoto Katori

    Journal of the Institute of Science and Engineering Chuo University   16 ( 16 )   1 - 20   2011.5

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  • 26pTC-13 Dunkl Intertwining Operator for the Noncolliding Brownian Motion

    Andraus Sergio, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   66 ( 1 )   313 - 313   2011.3

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  • 28aPS-74 Statistical Analysis for Temperature Fluctuations of Cosmic Microwave Background

    Nakahira R., Higuchi M., Kobayashi N., Kuninaka H., Yamazaki Y., Katori M., Matsushita M., Matsushita S., CHIANG Lung-Yih

    Meeting abstracts of the Physical Society of Japan   66 ( 1 )   364 - 364   2011.3

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  • 27pTF-2 Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background

    Kobayashi Naoki, Yamazaki Yoshihiro, Kuninaka Hiroto, Katori Makoto, Matsushita Mitsugu, Matsushita Satoki, Chiang Lung-Yih

    Meeting abstracts of the Physical Society of Japan   66 ( 1 )   333 - 333   2011.3

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  • 28aTB-11 Decomposition formula of the Schutz determinantal correlations in TASEP

    Fukazawa Tomohiro, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   66 ( 1 )   337 - 337   2011.3

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  • 27pTF-1 Fractal structures of statistical ensembles of random loops

    Nishikiori Yukina, Kobayasi Naoki, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   66 ( 1 )   332 - 332   2011.3

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  • 28aTB-10 Nonintersecting Systems of Loop-Erased Brownian Paths in the Plane

    Sato Makiko, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   66 ( 1 )   337 - 337   2011.3

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  • 24pTH-12 Dunkl Processes and Intertwining Operators

    Andraus Sergio, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   65 ( 2 )   247 - 247   2010.8

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  • 24pTH-9 Formin's determinants and nonintersecting loop-erased walks

    Sato Makiko, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   65 ( 2 )   247 - 247   2010.8

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  • 24pTH-10 Applicaton of Schutz Determinant on Semi-open TASEPs

    Fukazawa Tomohiro, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   65 ( 2 )   247 - 247   2010.8

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  • ランダム行列とその周辺

    香取眞理

    数理科学   48 ( 4 (No.562) )   13 - 19   2010.4

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  • 20aEJ-6 Random Walk-Driven Iterative Schwarz-Christoffel Transfomation

    Sato Fumihito, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   65 ( 1 )   270 - 270   2010.3

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  • ラプラス変換と揺らぎ

    香取眞理

    数理科学   48 ( 3(No.561) )   32 - 37   2010.3

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  • 連載(12)「問題例で深める物理」物理現象の次元性

    香取眞理, 中野徹

    数理科学   48 ( 1 (N0.559) )   59 - 65   2010.1

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  • 連載(11)「問題例で深める物理」流体力学からカオスへ

    香取眞理, 中野徹

    数理科学   47 ( 12 (No.558) )   66 - 71   2009.12

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  • 連載(10)「問題例で深める物理」多体系における集団運動と個別運動

    香取眞理, 中野徹

    数理科学   47 ( 11 (No.557) )   72 - 77   2009.11

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  • 連載(9)「問題例で深める物理」1次元イジング模型と転送行列

    香取眞理, 中野徹

    数理科学   47 ( 10 (No.556) )   72 - 77   2009.10

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  • 連載(8)「問題例で深める物理」変分原理

    香取眞理, 中野徹

    数理科学   47 ( 9 (No.555) )   65 - 70   2009.9

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  • 25aQC-13 Computer simulation of the Schramm-Loewner evolution and its applications

    Sato F., Katori M.

    Meeting abstracts of the Physical Society of Japan   64 ( 2 )   151 - 151   2009.8

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  • 連載(7)「問題例で深める物理」フーリエ級数とフーリエ変換

    香取眞理, 中野徹

    サイエンス社   47 ( 8 (No.554) )   72 - 77   2009.8

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  • 連載(6)「問題例で深める物理」磁場と電場の相対性

    香取眞理, 中野徹

    数理科学   47 ( 7 (No.553) )   72 - 77   2009.7

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  • 連載(5)「問題例で深める物理」マクスウェル方程式の微分形と積分形

    香取眞理, 中野徹

    数理科学   47 ( 6 (No.552) )   66 - 71   2009.6

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  • 連載(4)「問題例で深める物理」二体問題

    香取眞理, 中野徹

    数理科学   47 ( 5 (No.551) )   72 - 77   2009.5

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  • 連載「(3)問題例で深める物理」二項分布からカノニカル分布へ

    香取眞理, 中野徹

    数理科学   47 ( 4 (No.550) )   78 - 83   2009.4

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  • 30pRC-8 Limit Distributions of Quantum Walks and Ultraviolet Cutoff in Dirac Equations

    Sato F., Katori M.

    Meeting abstracts of the Physical Society of Japan   64 ( 1 )   342 - 342   2009.3

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  • 30pRC-6 Extreme Value Distributions of Noncolliding Brownian Paths

    Izumi Minami, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   64 ( 1 )   342 - 342   2009.3

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  • 30pRC-7 One-dimensional quantum random walks and the Gauss hypergeometric functions

    Ootani S., Katori M.

    Meeting abstracts of the Physical Society of Japan   64 ( 1 )   342 - 342   2009.3

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  • 連載(2)「問題例で深める物理」古典力学と量子力学の対応

    香取眞理, 中野徹

    数理科学   47 ( 3 (No.549) )   67 - 72   2009.3

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  • 連載(1)「問題例で深める物理」物理学の縦糸と横糸

    香取眞理, 中野徹

    数理科学   47 ( 2 (No.548) )   72 - 77   2009.2

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  • 「数学のノーベル賞」と統計物理学

    香取眞理

    数理科学   546 ( 12月号 )   5 - 6   2008.12

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  • 21pVC-13 Maximum Distributions of Noncolliding Bessel Bridges I

    Izumi M., Kobayashi N., Katori M.

    Meeting abstracts of the Physical Society of Japan   63 ( 2 )   224 - 224   2008.8

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  • Schramm-Loewner Evolution 入門

    香取眞理

    数理解析研究所講究録   1609   88 - 101   2008.7

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  • 25pPSA-45 Maximum value of the d-dimensional Bessel bridge

    Kobayashi Naoki, Izumi Minami, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   63 ( 1 )   326 - 326   2008.2

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  • 23aWE-1 Large Qudit Limit of One-dimensional Quantum Walks

    Sato M., Katori M.

    Meeting abstracts of the Physical Society of Japan   63 ( 1 )   265 - 265   2008.2

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  • 23aWE-2 Two-Dimensional Quantum Walk Models

    Watabe Kyohei, Katori Makoto

    Meeting abstracts of the Physical Society of Japan   63 ( 1 )   266 - 266   2008.2

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  • 23aWE-3 Two Bessel Bridges Conditioned Never to Collide and Multiple Dirichlet Series

    Izumi M., Kobayashi N., Katori M.

    Meeting abstracts of the Physical Society of Japan   63 ( 1 )   266 - 266   2008.2

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  • 対称性と調和と酔歩と・・・数学ワンダーランドへの誘い、砂田利一著「ダイヤモンドはなぜ美しい?離散調和解析入門」

    香取眞理

    数学セミナー   46 ( no.9/552 )   p.83   2007.9

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  • 臨界現象・フラクタル研究の新世紀--SLE の発見-- Invited Reviewed

    香取眞理

    日本物理学会誌   62 ( 7 )   527 - 531   2007.7

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  • 20aWA-10 Wigner formula of rotation matrices and quantum walks with multi-component qubits

    Miyazaki T., Katori M., Konno N.

    Meeting abstracts of the Physical Society of Japan   62 ( 1 )   278 - 278   2007.2

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  • ランダム行列から無限粒子系の数理へ

    香取眞理

    数理科学2007年2月号   No. 524   42 - 43   2007.2

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  • 土井正男, 統計力学, 朝倉書店, 東京, 2006, viii+227p, 21.5×15.5cm, 本体3,000円, (物理の考え方2), [学部向・大学院向]

    香取 眞理

    日本物理學會誌   61 ( 11 )   851 - 851   2006.11

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  • 新著紹介「統計力学」(土井正男著)

    香取眞理

    日本物理学会誌   61 ( 11 )   p.851   2006.11

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  • 経済物理の偶然と必然

    香取眞理

    数理科学2006年1月号   No.511   14 - 19   2006.1

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  • 物理学辞典(三訂版) Invited Reviewed

    物理学辞典編集委員会

    2005.9

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  • 21pYO-1 Quantum Walks and Weyl Equation

    Katori M., Fujino S., Konno N.

    Meeting abstracts of the Physical Society of Japan   60 ( 2 )   191 - 191   2005.8

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  • 21pYO-2 Orbital States of a Weyl Particle and Limit Theorem of Quantum Walks

    Fujino S., Katori M., Konno N.

    Meeting abstracts of the Physical Society of Japan   60 ( 2 )   191 - 191   2005.8

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  • Nonintersecting paths, noncolliding diffusion processes and representation theory

    Makoto Katori, Hideki Tanemura

    RIMS Kokyuroku   1438   83 - 102   2005.7

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    Other Link: http://hdl.handle.net/2433/47508

  • Non-colliding system of Brownian particles as Pfaffian process

    Makoto Katori

    RIMS Kokyuroku   1422   12 - 25   2005.4

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    Other Link: http://hdl.handle.net/2433/47202

  • 時間的に非斉次な非衝突ブラウン運動

    香取眞理, 種村秀紀

    立教 SFR 講究録 5 「ゲージ理論・行列模型と非平衡統計物理学」   5   36 - 76   2005.2

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  • 27aWM-9 Noncolliding diffusion processes and random matrix theories

    Katori M., Tanemura H.

    Meeting abstracts of the Physical Society of Japan   59 ( 1 )   266 - 266   2004.3

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  • 非衝突乱歩系・シューア関数・ランダム行列 Invited Reviewed

    香取眞理

    応用数理   13 ( 4 )   16 - 27   2003.12

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  • Mobius graphs expansion for moments of vicious walks

    Komatsuda N., Katori M.

    Meeting abstracts of the Physical Society of Japan   58 ( 1 )   297 - 297   2003.3

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  • Infinite systems of non-colliding Brownian particles

    Katori M., Tanemura H.

    Meeting abstracts of the Physical Society of Japan   58 ( 1 )   297 - 297   2003.3

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  • Dynamical correlation functions fbr vicious random walkers

    Nagao Taro, Katori Makoto, Tanemura Hideki

    Meeting abstracts of the Physical Society of Japan   57 ( 2 )   255 - 255   2002.8

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  • Genus expansion formula for moments of vicious walks

    Komatsuda N, Katori M

    Meeting abstracts of the Physical Society of Japan   57 ( 2 )   255 - 255   2002.8

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  • Scaling limit of vicious walks and two-matrix model

    M Katori, Tanemura H

    Meeting abstracts of the Physical Society of Japan   57 ( 2 )   254 - 254   2002.8

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  • Families of vicious walkers

    M Katori, Cardy J

    Meeting abstracts of the Physical Society of Japan   57 ( 2 )   255 - 255   2002.8

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  • Statistical laws in the income of Japanese companies

    T Mizuno, M Katori, H Takayasu, M Takayasu

    EMPIRICAL SCIENCE OF FINANCIAL FLUCTUATIONS   321 - 330   2002

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    Following the work of Okuyama, Takayasu and Takayasu [Okuyama, Takayasu and Takayasu 1999] we analyze huge databases of Japanese companies' financial figures and confirm that the Zipf's law, a power law distribution with the exponent -1, has been maintained over 30 years in the income distribution of Japanese companies with very high precision. Similar power laws are found not only in income distribution of company's income, but also in the distributions of capital, sales and number of employees.
    From the data we find an important time evolutionary property that the growth rate of income is approximately independent of the value of income, namely, small companies and large ones have similar statistical chances of growth. This observational fact suggests the applicability of the theory of multiplicative stochastic processes developed in statistical physics. We introduce a discrete version of Langevin equation with additive and multiplicative noises as a simple time evolution model of company's income. We test the validity of the TakayasuSato-Takayasu condition [Takayasu, Sato and Takayasu 1997] for having an asymptotic power law distribution as a unique statistically steady solution. Directly estimated power law exponents and theoretically evaluated ones are compared resulting a reasonable fit by introducing a normalization to reduce the effect of gross economic change.

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  • Random walk representation of the directed percolation

    Inui N., Katori M.

    Meeting abstracts of the Physical Society of Japan   56 ( 1 )   220 - 220   2001.3

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  • On the Drossel-Schwabl Limit in a Forest-Fire Model with Aging Trees

    Takamatsu T., Katori M.

    Meeting abstracts of the Physical Society of Japan   56 ( 1 )   219 - 219   2001.3

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  • Stochastic Laws in the Income of Japanese Companies and Randomly Amplified Langevin Equation II

    Mizuno T., Katori M., Takayasu H., Takayasu M.

    Meeting abstracts of the Physical Society of Japan   56 ( 1 )   273 - 273   2001.3

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  • Stochastic Laws in the Income of Japanese Companies and Randomly Amplified Langevin Equation

    Mizuno T., Katori M., Takayasu H., Takayasu M.

    Meeting abstracts of the Physical Society of Japan   55 ( 2 )   202 - 202   2000.9

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  • Statistical properties of the friendly walker

    Inui N., Katori M., Kameoka K.

    Meeting abstracts of the Physical Society of Japan   55 ( 2 )   196 - 196   2000.9

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  • Statistical and Stochastic Laws in the Income of Japanese Companies

    Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 2 )   237 - 237   2000.9

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  • Dissipative Abelian Sandpile Models and q=0 Potts Model

    Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 2 )   210 - 210   2000.9

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  • Drossel-Schwabl Limit and Self-Organized Criticality in Forest-Fire Model

    Takamatsu T., Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 2 )   196 - 196   2000.9

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  • Self-Organized Criticality of Forest-Fire Model with Aging Trees II

    Takamatsu T., Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 1 )   211 - 211   2000.3

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  • Extension of the Arrowsmith-Essam Theorem to the Domany-Kinzel Model

    Konno N., Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 1 )   203 - 203   2000.3

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  • Percolation and Wetting Transitions in Stochastic Models

    Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 1 )   204 - 204   2000.3

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  • Local and Global Survival Probabilities for the Domany-Kinzel Model

    Katori M., Konno N., Tanemura H.

    Meeting abstracts of the Physical Society of Japan   55 ( 1 )   204 - 204   2000.3

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  • Complete Convergence Theorem for the Domany-Kinzel Model

    Katori M., Konno N., Tanemura H.

    Meeting abstracts of the Physical Society of Japan   55 ( 1 )   204 - 204   2000.3

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  • Analysis of Canopy-Gap Structures of Forests by Thermal Equilibrium Correlation Equations

    Takamatsu T., Katori M.

    Meeting abstracts of the Physical Society of Japan   55 ( 1 )   212 - 212   2000.3

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  • J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Cambridge Univ. Press, Cambridge, 1999, 327+xv, 18×25.3cm, 本体17,820円, (Collection Alea-Sacley Monographs and Texts in Statistical Physics), [大学院向・専門書]

    香取 眞理

    日本物理學會誌   55 ( 2 )   132 - 133   2000.2

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  • 25aT-12 Self-Organized Criticality of Forest-Fire Model with Aging Trees

    TAKAMATSU T, KATORI M

    Meeting abstracts of the Physical Society of Japan   54 ( 2 )   217 - 217   1999.9

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  • 25aT-11 Structure of canopy in forest and surface of a certain random-field

    KIZAKI S, KATORI M

    Meeting abstracts of the Physical Society of Japan   54 ( 2 )   217 - 217   1999.9

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  • 28p-XD-15 Modeling of cooperative phenomena in phase transformation of locusts

    Kizaki S., Katori M.

    Meeting abstracts of the Physical Society of Japan   54 ( 1 )   648 - 648   1999.3

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  • 28p-XF-1 Critical behavior of three dimensional directed percolation

    Inui N., Kakuno H., Katori M., Komatsu G., Kameoka K.

    Meeting abstracts of the Physical Society of Japan   54 ( 1 )   653 - 653   1999.3

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  • 28p-XF-5 The size-distribution of avalanches for Abelian sandpile models II

    Shimamura M., Tsuchiya T., Katori M.

    Meeting abstracts of the Physical Society of Japan   54 ( 1 )   654 - 654   1999.3

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  • 28p-XF-4 Breaking of Self-Organized Criticality for Nonconservative BTW Models

    Tsuchiya T., Katori M., Shimamura M.

    Meeting abstracts of the Physical Society of Japan   54 ( 1 )   654 - 654   1999.3

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  • 北原和夫, 非平衡系の統計力学, 岩波書店, 東京, 1997, xvi+280p., 21.5×15cm, 本体3,500円 (岩波基礎物理シリーズ8) [学部向教科書]

    香取 眞理

    日本物理學會誌   53 ( 10 )   786 - 787   1998.10

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  • 北原和夫「非平衡系の統計力学」

    香取眞理

    日本物理学会誌   53 ( 10 )   786 - 787   1998.10

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  • Exact Solutions Directed Abelian Sandpile Models II

    TSUCHIYA T., KATORI M.

    Meeting abstracts of the Physical Society of Japan   53 ( 2 )   780 - 780   1998.9

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  • Forest Dynamics With Canopy Gap Expansion and Stochastic Ising Model

    KIZAKI S., KATORI M.

    Meeting abstracts of the Physical Society of Japan   53 ( 2 )   779 - 779   1998.9

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  • The size-distribution of avalanches for Abelian sandpile models

    SHIMAMURA M., TSUCHIYA T., KATORI M.

    Meeting abstracts of the Physical Society of Japan   53 ( 2 )   780 - 780   1998.9

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  • ワタリバッタ大発生のシナリオと確率モデル

    香取眞理, 木崎伸也

    Computer Today   85 ( 3 )   10 - 17   1998.5

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  • Exact Solutions for Directed Abelian Sandpile Model

    TSUCHIYA T., KATORI M.

    Meeting abstracts of the Physical Society of Japan   53 ( 1 )   650 - 650   1998.3

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  • 5p-M-7 Generalized sandpile models

    Kobayashi H., Katori M.

    Meeting abstracts of the Physical Society of Japan   52 ( 2 )   888 - 888   1997.9

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  • 6p-YE-14 Generating functions of the interacting random walks and correlation functions of the chiral Potts models

    Tsuchiya T, Katori M

    Meeting abstracts of the Physical Society of Japan   52 ( 2 )   760 - 760   1997.9

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  • 6p-YE-13 Gap distributions in rainforests studied by Ising models

    Katori M, Kizaki S, Terui Y

    Meeting abstracts of the Physical Society of Japan   52 ( 2 )   760 - 760   1997.9

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  • 30p-WE-2 Chiral Potts Model and Directed Percolation II

    Tsuchiya T., Katori M.

    Meeting abstracts of the Physical Society of Japan   52 ( 1 )   725 - 725   1997.3

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  • 30p-WE-1 Chiral Potts Model and Directed Percolation I

    Katori M., Tsuchiya T.

    Meeting abstracts of the Physical Society of Japan   52 ( 1 )   725 - 725   1997.3

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  • The contour method revisited -- Bounds for critical value of directed percolation --

    M.Katori, T.Tsuchiya, N.Satoh, S.Osanai, S.Kizaki

    中央大学理工学部紀要   39 ( 39 )   1 - 13   1997.3

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  • Series expansion for directed percolation and exclusive diffusion model

    M. Katori, H. Kobayashi, N. Inui, G. Komatsu, K. Kameoka

    Journal of the Institute of Science and Engineering, Chuo University   2   15 - 25   1997.3

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  • Mean-field approximations and Monte Carlo simulations for new sandpile models

    Koyayashi H., Katori M.

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1996 ( 3 )   594 - 594   1996.9

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  • Series expansion of asynmmetric directed percolation

    Inui Norio, Katori Makoto, Komatsu Gennichi, Kameoka Kouichi

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1996 ( 3 )   709 - 709   1996.9

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  • Ballot number representation for the series expansion of the directed percolation probability

    Katori M., Inui N.

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1996 ( 3 )   709 - 709   1996.9

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  • 田口善弘「砂時計の七不思議」-粉粒体の動力学-

    香取眞理

    中央評論   215   159 - 160   1996.4

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  • 31a-A-5 Catalan numbers is series expansion of the directed percolation

    Inui N., Katori M.

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   51 ( 3 )   544 - 544   1996.3

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  • 31a-A-6 Hypergeometric Series in a Series Expansion of the Directed-Bond Percolation Probability on the Square Lattice

    Katori M., Inui N.

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   51 ( 3 )   545 - 545   1996.3

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  • 31a-A-4 Critical exponent for wedge angle in bond directed percolation

    Tsukahara H., Katori M.

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   51 ( 3 )   544 - 544   1996.3

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  • Stochastic Cellular Automata and Directed Percolation

    M.Katori, H.Tsukahara, H.Kobayashi

    1   45 - 54   1996.3

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  • Cluster variational approach to non-equilibrium lattice models Invited Reviewed

    M Katori

    THEORY AND APPLICATIONS OF THE CLUSTER VARIATION AND PATH PROBABILITY METHODS   95 - 111   1996

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  • New mean-field approximation for the self-organized critical states in sandpile models

    Kobayashi H., Katori M.

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1995 ( 3 )   689 - 689   1995.9

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  • Spatio-Temporal Directed Percolation and its Continuous-Time Limit

    Tsukahara H., Katori M.

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1995 ( 3 )   731 - 731   1995.9

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  • Two-neighbour stochastic cellular automata and their planar lattice duals

    Katori M., Tsukahara H.

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1995 ( 3 )   731 - 731   1995.9

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  • Cluster approximations for the height distribution in the sandpile model

    Katori M., Kobayashi H.

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1995 ( 3 )   689 - 689   1995.9

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  • 一大学人と漫画

    香取眞理

    中央評論   212   53 - 57   1995.6

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  • 2-dimensional Directed Percolation and its Lattice Dual

    Tsukahara H, Katori M, Inui N

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   50 ( 3 )   624 - 624   1995.3

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  • Duality and Universality in Nonequilibrium Lattice Models

    Inui N, Katori M, Uzawa T

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   50 ( 3 )   625 - 625   1995.3

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  • 2p-E-1 Bernoulli property of the infinite totally asymmetric simple exclusion process

    KATORI Makoto

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1994 ( 3 )   444 - 444   1994.8

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  • 楠岡成雄著, 確率と確率過程, 岩波書店, 東京, 1993, x+92p., 21×15cm, 3,600円, 岩波講座応用数学2

    香取 眞理

    日本物理學會誌   49 ( 8 )   674 - 675   1994.8

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  • 楠岡成雄「確率と確率過程」

    香取眞理

    日本物理学会誌   49 ( 8 )   p.674   1994.8

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  • The Holley-Liggett argument revisited : an application to the generalized contact process

    M.Katori, N.Konno

    ICM94 Abstracts   p.149   1994.8

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  • 29p-C-4 Rigorous results fot the diffusive contact processes in d&ge;3

    KATORI Makoto

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   49 ( 3 )   539 - 539   1994.3

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  • 詳細釣合を満たさない定常分布:コンタクト・プロセスを例にして Invited Reviewed

    香取眞理

    日本物理学会誌   49 ( 2 )   100 - 107   1994.2

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    DOI: 10.11316/butsuri1946.49.100

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  • Phase Transitions in Contact Process and its Related Processes Invited Reviewed

    M.Katori, N.Konno

    Formation, Dynamics and statistics of Patterns ed. By K.Kawasaki and M.Suzuki   2   23 - 72   1993.12

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  • 13p-G-2 On Critical Values of Generalized Contact Processes in One Dimension II

    KATORI Makoto

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1993 ( 3 )   551 - 551   1993.9

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  • 14p-F-11 Structural and Statistical Properties of Directed Percolation

    ISOGAMI Sadao, KATORI Makoto, MATSUSHITA Mitsugu

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1993 ( 3 )   588 - 588   1993.9

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  • 13p-G-3 Existence of Phase Transition in the One-Dimensional Pair Annihilation Model

    KATORI Makoto, KONNO Norio

    Abstracts of the meeting of the Physical Society of Japan. Sectional meeting   1993 ( 3 )   551 - 551   1993.9

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  • 30p-F-8 Asymptotic behavior of diffusive contact process as D →+∞

    Konno Norio, Katori Makoto

    Abstracts of the meeting of the Physical Society of Japan. Annual meeting   48 ( 3 )   502 - 502   1993.3

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  • 27p-ZB-3 A Mean-Field Limit of Localization Transition

    Kawarabayashi T, Katori M, Suzuki M

    1992 ( 3 )   478 - 478   1992.9

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  • 28a-ZA-2 On Critical Values of Generalized Contact Processes in One Dimension

    KATORI Makoto, KONNO Norio

    1992 ( 3 )   514 - 514   1992.9

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  • 28a-ZA-1 Analysis on Critical Values for Diffusive Contact Processes

    KONNO Norio, KATORI Makoto

    1992 ( 3 )   514 - 514   1992.9

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  • 26p-ZB-3 Critical Phenomena of the S=1/2 Heisenberg-Ising Model Studied by CVM and CAM

    Tanaka Giichi, Kimura Minoru, Katori Makoto

    1992 ( 3 )   465 - 465   1992.9

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  • 30a-ZL-13 On Multiparticle Creation Processes

    KATORI Makoto, KONNO Norio

    47 ( 3 )   465 - 465   1992.3

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  • 27a-ZM-9 A Self- Consistent Approach to Mobility Edge Tranjectories of Localization

    Kawarabayasi T, Katori M, Suzuki M

    47 ( 3 )   387 - 387   1992.3

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  • 30a-ZL-12 Bounds on Critical Values for a Class of the Attractive NPSs with Finite Ranges

    KONNO Norio, KATORI Makoto

    47 ( 3 )   464 - 464   1992.3

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  • On Mathematical Models of Reaction-Diffusion Processes on Catalytic Surfaces

    M.Katori, N.Konno

    Pattern Formation in Complex Dissipative Systems ed. By S.Kai   171 - 175   1992.3

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  • On Mathematical Models of Reaction-Diffusion Processes on Catalytic Surfaces

    M.Katori, N.Konno

    Pattern Formation in Complex Dissipative Systems ed. By S.Kai   171 - 175   1992.3

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  • 拡散を伴ったコンタクト・プロセスについて(「非平衡系の統計物理」研究会(その2),研究会報告)

    香取 眞理, 今野 紀雄

    物性研究   59 ( 2 )   175 - 176   1992

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    Other Link: http://hdl.handle.net/2433/94986

  • 触媒表面の数理モデル(基研短期研究会「格子理論の進展-素粒子から生物まで-」,研究会報告)

    香取 眞理, 今野 紀雄

    物性研究   57 ( 6 )   775 - 779   1992

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    Other Link: http://hdl.handle.net/2433/94878

  • 無限粒子系の物理と数学(基研短期研究会『統計物理の現状と展望』~STATPHYS19に向けて~,研究会報告)

    香取 眞理

    物性研究   58 ( 5 )   545 - 552   1992

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    Other Link: http://hdl.handle.net/2433/94920

  • 28p-T-10 On Mathematical Models of Catalytic Surfaces

    KATORI Makoto, KONNO Norio

    46 ( 3 )   446 - 446   1991.9

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  • 28p-BPS-27 A Self-Consistent Approach to Localization in Disordered Electron Systems

    Kawarabayashi T., Katori M., Suzuki M.

    46 ( 3 )   462 - 462   1991.9

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  • 28p-T-11 Analysis of Order Parameter for a Class of the Uniform Nearest Particle Systems

    KONNO Norio, KATORI Makoto

    46 ( 3 )   446 - 446   1991.9

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  • 数学セミナー編集部編数学の最前線

    香取眞理

    日本物理学会誌   46 ( 8 )   p.700 - 700   1991.8

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  • 25p-F-5 Extinction of Branching Random Walk with Pair Annihilation

    KATORI Makoto, KONNO Norio

    1991 ( 3 )   430 - 430   1991.3

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  • セッション「無限粒子系の物理と数学」について(基研研究会「非可逆な多体系への統計物理及びその周辺分野からのアプローチ」報告,研究会報告)

    香取 眞理

    物性研究   57 ( 2 )   215 - 220   1991

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    Other Link: http://dl.ndl.go.jp/info:ndljp/pid/10936748

  • 非可逆な化学反応モデルにおける拡散の効果について(パターン形成、運動と統計,研究会報告)

    香取 眞理, 今野 紀雄

    物性研究   57 ( 3 )   397 - 400   1991

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    Other Link: http://hdl.handle.net/2433/94843

  • An Upper Bound for Survival Probability of Infected Region in the Contact Processes

    Makoto Katori, Norio Konno

    Journal of the Physical Society of Japan   60 ( 1 )   95 - 99   1991

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    An upper bound for the survival probability of infected region σ,λ(A), (A ⊂Zd), is presented for the contact process. The bound depends not only on the cardinality number |A| but also on the number of neighboring pairs of sites in A. It may correspond to Krinsky's bound using the Bethe approximation for the Ising models. The present result can be viewed as a generalization of Griffeath's bound for an order parameter ρ1,λ(≡ σ1,λ({χ})) in the one-dimensional system to the survival probabilities for all A in arbitrary dimensions. © 1991, THE PHYSICAL SOCIETY OF JAPAN. All rights reserved.

    DOI: 10.1143/JPSJ.60.95

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  • Phase Transition in Stationary States of Contact Process

    38 ( 2 )   243 - 256   1990.12

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  • 4p-PS-16 Applications of the Harris-FKG inequality to Upper Bounds for Order Parameters in the Contact Processes

    Konno N., Katori M.

    1990 ( 3 )   488 - 488   1990.10

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  • 4p-PS-17 Markov Extension and Upper Bounds for Survival Probability of the One-Dim.Contact Process

    Katori Makoto, Konno Norio

    1990 ( 3 )   488 - 488   1990.10

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  • Upper Bounds on Critical Value in the One-Dimensional Contact Process by a New Decoupling Procedure of Correlation Functions

    N.Konno, M.Katori

    ICM-90 Abstracts   p.164   1990.8

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  • Ta-no-ji approximation and CAM for the two dimensional square lattice

    Fujiki S., Katori M.

    45 ( 3 )   424 - 424   1990.3

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  • Upper and Lower Bounds for the Critical Value λ_c of Contact Process I

    Konno N., Katori M.

    45 ( 3 )   387 - 387   1990.3

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  • Applications of the Multi-Effective Field Theory and CAM to the 2D Ising models

    Minami K., Nonomura Y., Katori M., Suzuki M.

    45 ( 3 )   424 - 424   1990.3

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  • 14.コンタクト・プロセスにおける感染領域の定常分布ついて(「パターン形成、運動及びその統計」研究会,研究会報告)

    香取 眞理, 今野 紀雄

    物性研究   54 ( 4 )   310 - 312   1990

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  • 5a-R-9 Lower Bounds on the Critical Point in the Contact Process

    Katori M., Konno N.

    1989 ( 3 )   442 - 442   1989.9

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  • 5a-S-9 Applications of the Optimol Multi-Effective Field Theory to the Ising models

    Minami K., Nonomura Y., katori M., Suzuki M.

    1989 ( 3 )   452 - 452   1989.9

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  • 6p-R-7 Classification of stationary measures of 3-state Stochastic Cellular Automata

    Konno N., Katori M.

    1989 ( 3 )   478 - 478   1989.9

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  • C.J.Thompson Classical Equilibrium Statistical Mechanics

    44 ( 8 )   p.612 - 612   1989.8

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  • 28a-TB-5 CAM and The Series of the Approximations based on the Variational Method

    Kawashima N., Katori M., Suzuki M., Tsallis Constantino

    44 ( 3 )   334 - 334   1989.3

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  • 28a-TB-6 On the Cluster Variation Method and CAM Theory

    Fujiki Sumiyoshi, Katori Makoto

    44 ( 3 )   335 - 335   1989.3

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  • 31p-TC-12 Critical Phenomena of Epidemic Models

    Konno N., Katori M.

    44 ( 3 )   426 - 426   1989.3

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  • 31p-TC-11 Systematic Series of Approximations and the CAM Analysis

    Katori Makoto, Konno Norio

    44 ( 3 )   425 - 425   1989.3

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  • 18.Window-CID-CAMによるContact Processの研究(基研研究会「相転移研究の新手法とその応用」,研究会報告)

    香取 真理, 今野 紀雄

    物性研究   51 ( 5 )   465 - 469   1989

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    Other Link: http://hdl.handle.net/2433/93546

  • 2.カノニカル・シリーズの作り方 : クラスター平均場近似と相関等式切断近似(基研研究会「相転移研究の新手法とその応用」,研究会報告)

    香取 真理, 鈴木 増雄

    物性研究   51 ( 5 )   384 - 388   1989

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    Other Link: http://hdl.handle.net/2433/93561

  • 相転移とCAM理論(基研長期研究会「進化の力学への場の理論的アプローチ」,研究会報告)

    香取 真理, 鈴木 増雄

    素粒子論研究   78 ( 2 )   B200 - B204   1988.11

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  • 5p-D5-12 一次転移の極限としての2次転移 : ポッツ模型におけるコヒーレント異常と臨界現象

    香取 真理

    秋の分科会講演予稿集   1988 ( 3 )   407 - 407   1988.9

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  • 5p-D5-10 Contact Processの臨界現象

    今野 紀雄, 香取 眞理

    秋の分科会講演予稿集   1988 ( 3 )   406 - 406   1988.9

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  • 非平衡を捉える一つの視点

    香取眞理

    数理科学   301 ( 7 )   50 - 57   1988.7

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  • 相転移とCAM理論(基研長期研究会「進化の力学への場の理論的アプローチ」報告,研究会報告)

    香取 真理, 鈴木 増雄

    物性研究   51 ( 2 )   236 - 240   1988

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    Other Link: http://hdl.handle.net/2433/93496

  • 26a-X-10 クラスター変分法によるコヒーレント異常と臨界現象の研究 II

    香取 眞理, 鈴木 増雄

    秋の分科会講演予稿集   1987 ( 3 )   316 - 316   1987.9

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  • 26a-X-11 伝送行列を用いたコヒーレント異常法(CAM)とそのイジング模型への応用 III

    胡 暁, 香取 真理, 鈴木 増雄

    秋の分科会講演予稿集   1987 ( 3 )   317 - 317   1987.9

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  • 28a-QC-3 伝送行列を用いたコヒーレント異常法(CAM)とそのイジング模型への応用I

    胡 暁, 香取 真理, 鈴木 増雄

    秋の分科会講演予稿集   1986 ( 3 )   347 - 347   1986.9

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  • 28a-QC-2 コヒーレント異常法(CAM)による動的臨界現象の研究

    香取 眞理, 鈴木 増雄

    秋の分科会講演予稿集   1986 ( 3 )   346 - 346   1986.9

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  • GT線における非線形帯確率について(磁性体における新しいタイプの相転移現象,研究会報告)

    香取 眞理, 鈴木 増雄

    物性研究   46 ( 4 )   558 - 561   1986

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    Language:Japanese   Publisher:物性研究刊行会  

    CiNii Books

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    Other Link: http://dl.ndl.go.jp/info:ndljp/pid/10933559

  • 3a-C-2 2次元・3次元イジング模型における平均場有限サイズスケーリング理論

    香取 眞理, 鈴木 増雄

    秋の分科会講演予稿集   1985 ( 3 )   304 - 304   1985.9

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    Language:Japanese   Publisher:一般社団法人日本物理学会  

    CiNii Books

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  • レプリカ理論と非線形帯磁率(A.スピングラスの分子場理論と磁場効果,基研短期研究会「スピングラスとその周辺」,研究会報告)

    香取 真理, 鈴木 増雄

    物性研究   45 ( 2 )   107 - 111   1985

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    Language:Japanese   Publisher:物性研究刊行会  

    スピングラスの研究において,レプリカ理論はその平均場近似として多くの知見を与えている。ここでは,レプリカ理論を解説し,スピングラス相転移に特有な非線形帯磁率の異常について述べる。

    CiNii Books

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    Other Link: http://hdl.handle.net/2433/91865

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Presentations

  • Eigenvalues and pseudospectra of the non-Hermitian matrix-valued stochastic processes

    Makoto Katori

    2024.1 

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    Event date: 2024.1    

    Language:English   Presentation type:Oral presentation (general)   Country:Japan  

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  • An application of the theory of random point processes; Hyperuniformity of the bushes in deserts Invited

    Makoto Katori

    The 4th meeting of Grant-in-Aid for Transformative Research Areas `Establishing Data Descriptive Science and its Cross-Disciplinary Applications'  2024.1 

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    Event date: 2024.1    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

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  • Non-Hermitian matrix-valued Brownian motion and the regularized Fuglede-Kadison determinant random-fields Invited International conference

    Makoto Katori

    Workshop on Random Interacting Systems  ( Institute for Mathematical Sciences, National University of Singapore )   2023.12  Akira Sakai and Rongfeng Sun

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    Event date: 2023.12    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:Singapore  

    The non-Hermitian matrix-valued Brownian motion
    is the stochastic process of a random matrix
    whose entries are given by independent complex Brownian motions.
    The bi-orthogonality relation is imposed between
    the right and the left eigenvector processes, which allows for
    their scale transformations with an invariant eigenvalue process.
    The eigenvector-overlap process is a
    Hermitian matrix-valued process,
    each element of which is given by
    a product of an overlap of right eigenvectors
    and that of left eigenvectors.
    We derive a set of stochastic differential equations (SDEs)
    for the coupled system of the eigenvalue process and the
    eigenvector-overlap process and prove
    the scale-transformation invariance of the obtained SDE system.
    The Fuglede--Kadison (FK) determinant associated with the present
    matrix-valued process is regularized by introducing
    an auxiliary complex variable. This variable is necessary
    to give the stochastic partial differential equations (SPDEs)
    for the time-dependent random field defined by
    the regularized FK determinant
    and for its squared and logarithmic variations.
    Time-dependent point process of eigenvalues
    and its variation weighted by the diagonal elements of
    the eigenvector-overlap process are
    related to the derivatives of the
    logarithmic regularized FK-determinant random-field.
    We also discuss the PDEs obtained by averaging the SPDEs.
    The present talk is based on the joint work with
    Syota Esaki (Fukuoka) and Satoshi Yabuoku (Kitakyushu).
    A preprint is available at \url{https://arxiv.org/abs/2306.00300}.

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  • Non-Hermitian matrix-valued processes and the regularized Fuglede-Kadison determinant random-fields Invited

    Makoto Katori

    Random Operators and Related Topics  ( Hohoku University )   2023.10 

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    Event date: 2023.10    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:Japan  

    We consider a one-parameter family of
    non-Hermitian matrix-valued processes,
    which can be regarded as a dynamical extension
    of Girko's ensemble interpolating
    the GUE and the Ginibre ensemble of random matrices.
    For each process in this family,
    the bi-orthogonality relation is imposed between
    the right and the left eigenvector processes, which allows for
    the scale-transformation invariance of the system.
    There each eigenvalue process is coupled with
    the eigenvector-overlap process, which is a
    Hermitian matrix-valued process with entries given by
    products of overlaps of the right and left eigenvectors.
    The Fuglede--Kadison (FK) determinants of
    the present matrix-valued processes
    are regularized by introducing an auxiliary complex variable.
    Then, associated with the regularized FK determinants,
    the time-dependent random fields
    are defined in the two-dimensional complex space
    and their stochastic partial differential
    equations (SPDEs) are derived.
    Time-dependent point processes of eigenvalues
    and their variations weighted by the diagonal elements of
    the eigenvector-overlap processes are
    related to the logarithmic derivatives of the
    regularized FK-determinant random-fields.
    We also discuss the PDEs obtained by averaging the SPDEs.
    The present talk is based on the joint work with
    Yuya Tanaka, Saori Morimoto, Ayana Ezoe (Chuo),
    Syota Esaki (Fukuoka) and Satoshi Yabuoku (Kitakyushu).

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  • Particle systems for foraging ants and annihilation and creation of paths

    Saori Morimoto, Ayana Ezoe, Yuya Tanaka, Makoto Katori, Hiraku Nishimori

    2023 Annual Meeting of the Japan Society for Mathematical Biology  ( Nara Women's Univarsity )   2023.9  The Japan Society for Mathematical Biology

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    Event date: 2023.9    

    Language:Japanese   Presentation type:Oral presentation (general)   Country:Japan  

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  • Switching particle systems for foraging ants and phase transitions in path selection

    Ayana Ezoe, Saori Morimoto, Yuya Tanaka, Makoto Katori, Hiraku Nishimori

    2023 Annual Meeting of the Japan Society for Mathematical Biology  ( Nara Women's Univarsity )   2023.9  The Japan Society for Mathematical Biology

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    Event date: 2023.9    

    Language:Japanese   Presentation type:Oral presentation (general)   Country:Japan  

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  • Stochastic log-gases, multiple SLEs, and Gaussian free fields Invited International conference

    Makoto Katori

    New trends of conformal theory from probability to gravity  ( Okinawa Institute of Science and Technology (OIST) )   2023.7  Nicolas Delporte, Reiko Toriumi, Shinobu Hikami (OIST)

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    Event date: 2023.7 - 2023.8

    Language:English   Presentation type:Oral presentation (invited, special)   Country:Japan  

    Multiple extensions of Schramm--Loewner evolution (SLE)
    have been usually studied by the method using
    commutation relations of SLEs and the absolute continuity of measures.
    This method is, however, available only in the case such that
    the curves generated by commuting SLEs are separated from each other.
    Recently we have found that this restriction can be removed if
    we impose the coupling between the multiple SLEs and the appropriately
    modified Gaussian free fields (GFFs), where we have followed the
    work by Dub\'edat, Sheffield, Miller, and others on the coupling
    of a single SLE and GFF.
    By It\^o's stochastic calculus, we prove that proper couplings
    between multiple SLEs and GFFs are established if and only if
    the driving processes of multiple SLEs are of
    a specified type of interacting particle systems.
    They are identified with the log-gases extensively studied as
    dynamical extensions of the eigenvalue distributions
    in random matrix ensembles.
    We will show that this new connection between SLE and
    random matrix theory (RMT) enables us to enjoy variety of
    coupled systems due to the variety of ensembles in RMT.
    The law of large numbers are then clarified
    as the Wigner semicircle law and its variations in the RMT level
    and as the Loewner chains driven by
    the processes solving the Burgers-type PDEs
    in the SLE level.
    The present talk is based on the joint work with
    Shinji Koshida (Aalto University).

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  • Elliptic Extensions of Noncolliding Brownian Bridges Invited International conference

    Makoto Katori

    Processes and heat kernels with symmetries (Conference with mini-courses)20  ( Angers, France )   2023.6  Piotr Graczyk (Université d'Angers), Alessandra Occelli (Université d’Angers), Patrice Sawyer (Laurentian University, Sudbury)

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    Event date: 2023.6    

    Language:English   Presentation type:Poster presentation   Country:France  

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  • PDE of the potential field for eigenvalues and eigenvector-overlaps of the non-Hermitian matrix-valued BM International coauthorship International conference

    Yuya TANAKA, Saori MORIMOTO, Ayana EZOE, Makoto KATORI

    Random Matrices and Applications  ( RIMS, Kyoto University )   2023.6  Organaizers: Benoit Collins(Kyoto), Catherine Donati-Martin(Versailles), Noriyoshi Sakuma(Nagoya City), Tomoyuki Shirai(Kyusyu), Ofer Zeitouni(Weizmann Institute of Science)

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    Event date: 2023.6    

    Language:English   Presentation type:Poster presentation   Country:Japan  

    The non-Hermitian matrix-valued Brownian motion (BM) induces
    a coupled system of the eigenvalue process
    and the eigenvector-overlap process.
    Here the eigenvector-overlap process
    is a matrix-valued process, each element of which
    is given by a product of an overlap of right eigenvectors
    and that of left eigenvectors.
    We consider
    the time-dependent random field defined on the two-dimensional
    complex space using the notion of
    the reguralized Fuglede--Kadison determinant
    of the matrix-valued stochastic process.
    As explained in a talk by one of the present authors (MK),
    this random field plays an important role for the system.
    In this presentation, we take average of this random field
    to define a deterministic field and determine the
    partial differential equation (PDE) which this field satisfies.
    We call this field the potential field,
    since its derivatives give
    the time-dependent density functions of the eigenvalue process
    and the eigenvector-overlap process.
    The present consideration will provide a mathematical interpretation
    of the calculus reported in physics literature
    (e.g., Burda et al. (2015)).
    The exact solutions of the PDE
    and explicit expressions for the density functions
    are shown for several initial conditions.
    Comparison with numerical simulations will be
    also reported.

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  • Eienvalues, eigenvector-overlaps, and regularized Fuglede-Kadison determinant of the non-Hermitian matrix-valued Brownian motion Invited International coauthorship International conference

    Makoto Katori

    Random Matrices and Applications  ( RIMS, Kyoto University )   2023.6  Organaizers: Benoit Collins(Kyoto), Catherine Donati-Martin(Versailles), Noriyoshi Sakuma(Nagoya City), Tomoyuki Shirai(Kyusyu), Ofer Zeitouni(Weizmann Institute of Science)

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    Event date: 2023.6    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:Japan  

    The non-Hermitian matrix-valued Brownian motion
    is the stochastic process of a random matrix
    whose entries are given by independent complex Brownian motions.
    The bi-orthogonality relation is imposed between
    the right and the left eigenvector processes, which allows for
    their scale transformations with an invariant eigenvalue process.
    The eigenvector-overlap process is a matrix-valued process,
    each element of which is given by
    a product of an overlap of right eigenvectors
    and that of left eigenvectors.
    We derive a set of stochastic differential equations
    for the coupled system of the eigenvalue process and the
    eigenvector-overlap process and prove
    the scale-transformation invariance of the system.
    The Fuglede--Kadison (FK) determinant associated with the present
    matrix-valued process is regularized by introducing
    an auxiliary complex variable. This variable is necessary
    to give the stochastic partial differential equations (SPDEs)
    for the time-dependent random field associated with
    the regularized FK determinant and for its logarithmic variation.
    Time-dependent point process of eigenvalues
    and its variation weighted by the diagonal elements of
    the eigenvector-overlap process are
    related to the derivatives of the
    logarithmic random-field of the regularized FK determinant.
    From the SPDEs, a system of PDEs
    for the density functions of these two types of
    time-dependent point processes are obtained.
    The present talk is based on the joint work with
    Syota Esaki (Fukuoka) and Satoshi Yabuoku (Kitakyushu).

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  • Elliptic extensions in statistical and stochastic systems Invited International conference

    Makoto Katori

    International Conference on Mathematical Methods in Physics  ( Marrakesh, Morocco )   2023.4  H. Hedenmalm. Z. Mouayn, P. Cerejeiras, S. N. Lagmiri

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    Event date: 2023.4    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:Morocco  

    I will report the recent progress
    of the elliptic extensions in the study of statistical and
    stochastic models in
    equilibrium and nonequilibrium statistical mechanics and
    probability theory $[1]$.
    Using the $R_n$ theta functions $\{\psi^{R_n}_j\}_{j=1}^n$, $n \in \mathbb{N}$
    introduced by Rosengren and Schlosser (2006)
    in association with the seven irreducible reduced affine
    root systems,
    $R_n=A_{n-1}$, $B_n$, $B^{\vee}_n$,
    $C_n$, $C^{\vee}_n$, $BC_n$, $D_n$,
    and their extensions denoted by $\{\Psi^{R_n}_j\}_{j=1}^n, n \in \mathbb{N}$,
    we discuss two groups of
    interacting particle systems, in which $n$ indicates
    the total number of particles in each system
    and $R$ specifies one of the seven types for each model.
    The first group of systems describes
    the noncolliding Brownian bridges
    on a one-dimensional torus $\mathbb{T}$ (\textit{i.e.}, a circle) or an interval,
    and the second one the determinantal point processes
    (DPPs) on a two-dimensional torus $\mathbb{T}^2$.
    The former systems are $(1+1)$-dimensional stochastic processes
    and the latter ones are statistical models of
    stationary point configurations in two dimensions,
    both of which provide mathematical models for
    systems of fermions,
    which are interacting with repulsive forces.
    The boundary conditions
    and the initial/final configurations
    of the noncolliding Brownian bridges,
    and the periodicity conditions of the DPPs are systematically
    changed depending on the choice of $R$ from the seven types.
    We argue the scaling limits
    associated with $n \to \infty$ and precisely define
    the infinite particle systems.
    The connections
    to other statistical systems
    such as the one-component plasma models
    and the Gaussian free fields will be also discussed. \\
    {[1]} Katori, M.: \textit{Elliptic Extensions in Statistical and Stochastic
    Systems}, SpringerBriefs in Mathematical Physics 47,
    Springer (2023).

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  • Stochastic processes on a complex plane associated with non-Hermitian matrix-valued Brownian motions

    Makoto Katori

    2023.1 

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    Event date: 2023.1    

    Language:Japanese   Presentation type:Oral presentation (general)   Country:Japan  

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  • Two-dimensional processes associated with the non-Hermitian matrix-valued Brownian motions Invited

    Makoto Katori

    Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields 2023  ( Ni )   2023.1  Japan-Netherlands Research Cooperative Program between JSPS and NWO

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    Event date: 2023.1    

    Language:English   Presentation type:Oral presentation (invited, special)  

    It is well known that the Hermitian matrix-valued Brownian motion (BM),
    H(t), is decoupled into the unitary matrix-valued process
    diagonarlizing H(t) and obtained eigenvalue process.
    The latter process is identified with the Dyson BM with a special
    value of parameter and is realized as the non-colliding BM
    in one dimension.
    It is regarded as the dynamical extension of GUE
    studied in random matrix theory (RMT).
    In the present talk, we discuss the non-Hermitian matrix-valued BM,
    M(t), whose eigenvalues move on the complex plane
    and exhibit a dynamics of the Ginibre ensemble studied in RMT.
    We impose the bi-orthonormality condition between the left- and
    the right-eigenvectors at each time. This means that the
    matrix-valued process diagonalizing M(t) is non-unitary.
    First I will show movies of computer simulations provided by
    my students, Yuya Tanaka, Saori Morimoto, and Ayana Ezoe,
    in order to demonstrate the coupling phenomena
    of the eigenvalue process and the non-unitary matrix-valued
    process for a variety of initial conditions.
    Then I will briefly review the recent results on the SDE
    representation of the processes obtained by Esaki and Yabuoku.
    In contrast with the Dyson BM having the repulsive
    interaction between any pair of particles,
    the eigenvalue process of M(t) is martingale.
    The cross-variations of the eigenvalues define a
    matrix-valued process. It exhibits a time-evolution of
    the overlap matrix whose statistics was
    first studied by Chalker and Mehlig
    for the Ginibre ensemble in RMT.
    We introduce the time-dependent point processes
    of eigenvalues weighted by the values of overlap-matrix elements,
    in addition to the usual eigenvalue point-process.
    Following the electrostatic analogy reported by
    Burda et al. (2015) in physics literature,
    possible hydrodynamic descriptions of these processes
    on the complex plane are shown.

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  • On non-Hermitian matrix-valued Brownian motions Invited

    Makoto Katori

    The 20th Symposium Stochastic Analysis on Large Scale Interacting Systems  ( Nishijin Plaza, Kyushu University )   2022.12 

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    Event date: 2022.12    

    Language:Japanese   Country:Japan  

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  • Multiple Schramm--Loewner evolution and Dyson's Brownian motion model Invited

    Makoto Katori

    Mathematical Society of Japan 2022 Autumn Meeting  ( Hokkaido University )   2022.9  Mathematical Society of Japan

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    Event date: 2022.9    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

    Schramm--Loewner evolution (SLE) is a stochastic extension
    of the classical Loewner theory in complex analysis
    on a simply connected proper domain of ${\mathbb{C } }$
    such that the driving function of the Loewner chain of
    conformal maps is given by a one-dimensional stochastic process.
    Here we assume that the domain is the upper-half
    complex-plane ${\mathbb{H } }$ and the driving process runs
    on ${\mathbb{R } }$.
    It was proved that the SLE driven by a time changed Brownian
    motion on ${\mathbb{R } }$, $(B_{\kappa t})_{t \geq 0}$, $\kappa >0$,
    determines a one-parameter family of probability laws
    of a random continuous curve in $\overline{\mathbb{H } }$
    connecting 0 and $\infty$,
    and that this family, denoted by
    SLE$_{\kappa}$, $\kappa >0$, covers all probability laws of such
    curves having the conformal invariance and
    the domain Markov property.
    In probability theory and statistical physics,
    the SLE$_{\kappa}$ has been playing a central role in studying
    critical phenomena associated with continuous phase transitions
    and fractal geometries in two dimensions.
    Therefore, it is natural to generalize the SLE theory to describe
    random multiple curves in ${\mathbb{H } }$ driven by
    an interacting particle system on ${\mathbb{R } }$.
    The problem is that the requirement of the conformal invariance
    and the domain Markov property is not sufficient to determine
    a `canonical' family of multiple SLEs.
    Motivated by the recent work by Sheffield on the
    quantum gravity zipper
    and a series of papers by Miller and Sheffield
    on the imaginary geometry,
    we employ the coupling of Gaussian free fields (GFFs)
    and multiple SLEs.
    We show that
    a multiple SLE is correctly coupled with a certain GFF
    if and only if the driving particle system is
    given by Dyson's Brownian motion model studied
    in random matrix theory.
    Dyson's Brownian motion model is also a one-parameter
    ($\beta>0$) family of one-dimensional log-gases.
    The coupling is achieved if and only if $\beta=8/\kappa$.
    The present study on the GFF/multiple SLE coupling enables
    us to clarify the basic properties of the constructed multiple SLE
    (e.g., continuity of multiple SLE curves,
    `phase transitions' at $\kappa=4$ and 8). Moreover, we expect that
    a notion of conformal invariance of SLE
    will enrich the study of random matrix theory
    via the present results on the trinity of
    the GFF, the multiple SLE, and Dyson's Brownian motion model.
    The present talk is based on a joint work
    with Shinji Koshida (Aalto University).

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  • Random Matrices, Multiple SLE, and Quantum Gravity Invited

    Makoto Katori

    Summer School Mathematical Physics 2022, K. Itô meets M. Sato  ( Graduate School of Mathematical Sciences, the University of Tokyo )   2022.8  Graduate School of Mathematical Sciences, the University of Tokyo, Professor Y. Ogata and Professor Y. Kawahigashi

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    Event date: 2022.8    

    Language:Japanese   Presentation type:Public lecture, seminar, tutorial, course, or other speech   Country:Japan  

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  • Point processes and multiple-SLE/GFF coupling Invited International conference

    Makoto Katori

    Fourth ZiF Summer School, Randomness in Physics and Mathematics, From Integrable Probability to Disordered Systems  ( ZiF - Zentrum für interdisziplinäre Forschung, Universität Bielefeld, Bielefeld, Germany )   2022.8  ZiF - Zentrum für interdisziplinäre Forschung, Universität Bielefeld, Bielefeld, Germany

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    Event date: 2022.8    

    Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech   Country:Germany  

    (i) a variety of determinantal point processes (DPPs), and
    (ii) the Gaussian analytic functions (GAFs) on a disk and an annulus, and point
    processes given by zeros of the GAFs.
    (iii) the Brownian motion, the Bessel process, the Schramm-Loewner evolution
    (SLE), Dyson’s Brownian motion,
    (iv) the Gaussian free field (GFF) and the coupling between GFF and the multiple
    SLE driven by Dyson’s Brownian motion. Notice that Dyson’s Brownian motion
    with beta=2 can be regarded as a dynamical version of DPP.

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  • Matrix-valued Brownian motions and two-dimensional processes Invited International conference

    Makoto Katori

    Workshop `Crossroad of Statistical Physics and Probability Theory’  ( Korakuen Campus, Chuo University, )   2022.6 

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    Event date: 2022.6    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

    I have studied Dyson's Brownian motion model with
    Hideki Tanemura and others, which
    is a Hermitian matrix-valued Brownian motion.
    The eigenvalue process is determinantal in space and time
    and the hydrodynamic limit is described by
    the complex Burgers equation in the inviscid limit.
    In the present talk, I review the recent work by
    Burda et al.~(2015), Grela and Warcho\l~(2018),
    Bourgade and Dubach (2020),
    Akemann et al.~(2020), and others
    on a non-Hermitian matrix-valued Brownian motion.
    I will explain new aspects of its eigenvalue process on the
    complex plane: It is dynamically coupled with
    the left and right eigenvectors and hence
    essentially different from Dyson's process on
    the real line.
    Nevertheless, the inviscid Burgers equation
    and the determinantal structure appear
    associated with the eigenvector processes.
    Some numerical results obtained by my students
    will be also demonstrated.

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  • Random point processes (random matrix theory), random curves (SLE), and random surfaces (quantum gravity) Invited

    Makoto Katori

    9th Meeting of Statistical Physics  ( online )   2022.3  K. Saito (Keio Univ.), H. Tasaki (Gakusyuin Univ.)

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    Event date: 2022.3    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

    File: Katori_StatPhys_March2022_v5.pdf

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  • Bosonic fields and fermionic processes Invited International conference

    Makoto Katori

    Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields 2022  ( Kyushu University、Ito Campus )   2022.1 

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    Event date: 2022.1    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

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  • Gaussian analytic functions and permanental-determinantal point processes on an annulus Invited

    Makoto Katori

    Tokyo Probability Seminar  2021.12 

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    Event date: 2021.12    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

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  • Hyperuniformity of the determinantal point processes associated with the Heisenberg group Invited

    Makoto Katori

    Mathematical Aspects of Quantum Fields and Related Topics  ( Kyoto (Zoom) )   2021.12  Research Institute for Mathematical Sciences, Kyoto University

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    Event date: 2021.12    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

    The \textit{Ginibre point process} is given by the eigenvalue
    distribution of a non-hermitian complex Gaussian matrix
    in the infinite matrix-size limit.
    This is a \textit{determinantal point process} (DPP) on the complex
    plane $\C$ in the sense that all correlation functions
    are given by determinants specified by an integral
    kernel called the \textit{correlation kernel}.
    As extensions of the Ginibre DPP,
    a one-parameter family of DPPs is defined on
    $\C^d$, in which $d \in \N$ is the parameter
    and the Ginibre DPP is regarded as
    the lowest-dimensional case ($d=1$).
    We call it the \textit{Heisenberg family of DPPs}, and
    we will explain this reason in the first part of the present talk:
    For each $d \in \N$, the correlation kernel is expressed by
    a matrix-element function of the
    \textit{Schr\"{o}dinger representation}
    of the \textit{Heisenberg group}, and is identified with
    the \textit{reproducing kernel}
    of the \textit{Bargmann-Fock space}.
    Then we prove that any DPP in this family has
    \textit{hyperunifomity} as follows:
    We consider a series of bounded domains
    $\Lambda_n \subset \C^d$,
    which are monotonically increasing and
    $\Lambda_n \to \C^d$ as $n \to \infty$.
    They provide a series of `observation windows' to measure
    local density-fluctuation for an infinite point process
    $\Xi$, and then the hyperuniformity is defined by
    \[
    \lim_{n \to \infty} \frac{\var[\Xi(\Lambda_n)]}{\bE[\Xi(\Lambda_n)]}=0.
    \]
    We show that when $\Lambda_n$ are polydisks, the proof is
    readily given by using the \textit{duality relation}
    between DPPs.
    In the case that $\Lambda_n$ are balls, we have derived
    exact formulas of $\var[\Xi(\Lambda_n)]$ using the modified
    Bessel functions, which provide asymptotic expansions
    for $\var[\Xi(\Lambda_n)]$ in $\vol[\Lambda_n] \to \infty$.
    This talk is based on the joint work with
    T. Matsui (Chuo Univ.) and T. Shirai (Kyushu Univ.),
    and with S. Koshida (Aalto Univ.).

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  • Orthogonal theta functions associated with affine root systems and determinantal point processes Invited International conference

    Makoto Katori

    CIRM Conference - Modern Analysis Related to Root Systems with Applications  ( Luminy, Marseille, France (hybrid) )   2021.10  Centre International de Rencontres Mathematiques

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    Event date: 2021.10    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:France  

    Associated with the seven families of irreducible reduced
    affine root systems $R$,
    Rosengren and Schlosser introduced the families of
    $R$-theta functions.
    We consider appropriate inner products and construct
    the orthonormal bases for the Hilbert spaces of $R$-theta functions.
    Using them the reproducing kernels are defined
    and the families of determinantal point processes (DPPs)
    with finite numbers of particles are obtained on a complex plane.
    There the correlation kernels are given by the
    reproducing kernels.
    The double periodicity and symmetry of the DPPs are studied.
    Infinite particle limits provide the Ginibre DPP
    well studied in random matrix theory
    as well as new kinds of infinite DPPs.

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    Other Link: https://www.cirm-math.fr/Schedule/screen_display.php?id_renc=2404

  • Zeros of the i.i.d. Gaussian Laurent series on an annulus Invited International conference

    Makoto Katori

    Stochastic Differential Geometry and Mathematical Physics  ( Rennes (online) )   2021.6  The Henri Lebesgue Center

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    Event date: 2021.6    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:France  

    On an annulus
    ${\mathbb{A } }_q :=\{z \in {\mathbb{C } }: q < |z| < 1\}$
    with a fixed $q \in (0, 1)$,
    we study a Gaussian analytic function (GAF)
    defined by the
    i.i.d.~Gaussian Laurent series.
    The covariance kernel of the GAF is given by
    the weighted Szeg\H{o} kernel of ${\mathbb{A } }_q$
    with the weight parameter $r$
    studied by Mccullough and Shen.
    Conditioning the GAF by giving zeros,
    new GAFs are induced such that
    the covariance kernels are also given by
    the weighted Szeg\H{o} kernel of Mccullough and Shen
    but the weight parameter $r$
    is changed depending on the given zeros.
    We prove that the zero set of the GAF provides
    a permanental-determinantal point process (PDPP)
    in which each correlation function is expressed by
    a permanent multiplied by a determinant.
    If we take the limit $q \to 0$,
    a simpler but still non-trivial
    PDPP is obtained on a punctured unit disk
    ${\mathbb{D } }^{\times} := {\mathbb{D } } \setminus \{0\}$.
    In the further limit $r \to 0$
    the present PDPP is reduced to the
    determinantal point process on ${\mathbb{D } }$ studied by
    Peres and Vir\'ag.
    The present talk is based on
    a joint work with Tomoyuki Shirai
    (https://arxiv.org/abs/2008.04177).

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  • Complex Burgers equations for log-gases and free convolutions Invited

    Makoto Katori

    Workshop on `Random matrices, Determinantal point processes and Gaussian analytic functions'  ( online )   2021.3  Kyushu University, Institute of Math for Industry

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    Event date: 2021.3    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

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  • Gaussian analytic function with the Szego kernel on an annulus and its zeros Invited

    Makoto Katori

    Department of Mathematics Colloquium, Kyoto University  ( online )   2020.10  Department of Mathematics, Kyoto University

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    Event date: 2020.10    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

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  • Elliptic extensions of determinantal point processes Invited

    Makoto Katori

    Mathematical Society of Japan, Autumn Meeting  ( Kumamoto University (online) )   2020.9  Mathematical Society of Japan

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    Event date: 2020.9    

    Language:Japanese   Presentation type:Oral presentation (invited, special)   Country:Japan  

    A point process is a statistical ensemble of
    random nonnegative-integer-valued Radon measures
    on a space equipped with a reference measure.
    We consider the case in which for any integer $n$
    an $n$-point correlation function is well defined
    with respect to the $n$-product of reference measure.
    When an $n$-point correlation function is given by
    a determinant of $n \times n$ matrix for every $n$
    and the entries of matrices are
    determined by a kernel of an integral operator,
    the point process is said to be determinantal
    and the kernel is called the correlation kernel.
    Many examples of determinantal point processes (DPPs)
    have been studied in random matrix theory and correlation
    kernels provide reproducing kernels
    which construct reproducing kernel Hilbert spaces.
    Recently we are interested in the DPPs
    in which correlation kernels are expressed using
    Jacobi's theta functions and Weierstrass' elliptic functions.
    In the present talk we will explain that
    these new examples of DPPs can be regarded as
    elliptic extensions of the classical DPPs and their
    $q$-extensions (trigonometric extensions).
    In particular we will report an elliptic extension of the
    beautiful work by Peres and Vir\'ag published in 2005,
    who considered the Gaussian analytic function (GAF)
    on a unit disk ${\mathbb{D } }$ defined as
    an ensemble of random power series with i.i.d.~complex
    Gaussian coefficients.
    There the covariance kernel of GAF is
    given by the reproducing kernel of the Hardy space on
    ${\mathbb{D } }$ called the Szeg\H{o} kernel of ${\mathbb{D } }$.
    They showed that zeros of the GAF form a DPP whose
    correlation kernel is equal to the Bergman kernel
    of ${\mathbb{D } }$.
    This talk is based on the joint work with Tomoyuki Shirai
    (IMI, Kyushu University).

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  • Determinantal Point Processes, Stochastic Log-Gases, and Beyond Invited International conference

    Makoto Katori

    Workshop `Probability and Stochastic Processes’  ( Orange County, Coorg )   2020.2  the Indian Academy of Sciences

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    Event date: 2020.2    

    Language:English   Presentation type:Oral presentation (invited, special)   Country:India  

    A determinantal point process (DPP) is an ensemble
    of random nonnegative-integer-valued Radon measures,
    whose correlation functions are all given by determinants
    specified by an integral kernel called the correlation kernel.
    First we show our new scheme of DPPs in which
    a notion of partial isometies between a pair of Hilbert spaces
    plays an important role.
    Many examples of DPPs in one-, two-, and higher-dimensional
    spaces are demonstrated, where several types
    of weak convergence from finite DPPs to
    infinite DPPs are given.
    Dynamical extensions of DPP are realized in one-dimensional
    systems of diffusive particles conditioned
    never to collide with each other.
    They are regarded as one-dimensional stochastic log-gases,
    or the two-dimensional Coulomb gases
    confined in one-dimensional spaces.
    In the second section, we consider such
    interacting particle systems in one dimension.
    We introduce a notion of
    determinantal martingale and prove that,
    if the system has determinantal martingale representation (DMR),
    then it is a determinantal stochastic process (DSP) in the sense that
    all spatio-temporal correlation function are expressed by
    a determinant.
    In the last section, we construct processes of
    Gaussian free fields (GFFs) on simply connected
    proper subdomains of $\C$ coupled with
    interacting particle systems defined
    on boundaries of the domains.
    There we use multiple Schramm--Loewner evolutions (SLEs)
    driven by the interacting particle systems.
    We prove that, if the driving processes
    are time-changes of the log-gases studied in the
    second section, then the obtained GFF with multiple SLEs
    are stationary.
    The stationarity defines an equivalence relation of
    GFFs, which will be regarded as a generalization of
    the imaginary surface studied by Miller and Sheffield.

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  • Correlation kernels of determinantal point processes expressed by Jacobi theta functions

    Makoto Katori

    2020.1 

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    Event date: 2020.1    

    Language:Japanese   Presentation type:Oral presentation (general)   Country:Japan  

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  • Zeros of Gaussian analytic functions in the annulus and hyperdeterminantal point processes Invited

    KATORI Makoto

    Spectra of Random Operators and Related Topics  ( Gakushuin University )   2020.1 

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  • Gaussian free fields coupled with stochastic log-gases via multiple SLEs Invited

    KATORI Makoto

    Stochastic Analysis on Particle Systems  ( Keio University, Hiyoshi Campus )   2019.12 

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  • Zeros of the i.i.d. Gaussian Laurent series in the annulus Invited

    KATORI Makoto

    The 18th Symposium `Stochastic Analysis on Large Scale Interacting Systems'  ( Department of Mathematics, Osaka University )   2019.11 

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    File: schedule.pdf

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  • Partial isometries, duality, and determinantal point processes Invited International conference

    KATORI Makoto

    Japanese-German Open Conference on Stochastic Analysis 2019  ( Fukuoka University )   2019.9  Japanese-German Open Conference

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    Language:English   Presentation type:Oral presentation (invited, special)  

    A determinantal point process (DPP) is
    an ensemble of random nonnegative-integer-valued
    Radon measures $\Xi$ on
    a space $S$ with measure $\lambda$,
    whose correlation functions are all
    given by determinants specified by an
    integral kernel $K$ called the correlation kernel.
    We consider a pair of Hilbert spaces,
    $H_{\ell}, \ell=1,2$, which are assumed to
    be realized as $L^2$-spaces,
    $L^2(S_{\ell}, \lambda_{\ell})$, $\ell=1,2$,
    and introduce
    a bounded linear operator ${\cal W} : H_1 \to H_2$
    and its adjoint ${\cal W}^{\ast} : H_2 \to H_1$.
    We prove that if both of ${\cal W}$ and ${\cal W}^{\ast}$ are
    partial isometries and both of ${\cal W}^{\ast} {\cal W}$ and
    ${\cal W} {\cal W}^{\ast}$
    are of locally trace class, then we have unique pair
    of DPPs, $(\Xi_{\ell}, K_{\ell}, \lambda_{\ell})$, $\ell=1,2$,
    which satisfy useful duality relations.
    We assume that ${\cal W}$ admits an integral kernel $W$
    on $L^2(S_1, \lambda_1)$,
    and give practical setting of $W$ which makes
    ${\cal W}$ and ${\cal W}^{\ast}$ satisfy the above conditions.
    By showing several examples,
    we demonstrate that the class of DPPs
    obtained by our method is large enough to
    study universal structures of DPPs.
    The present talk is based on
    a joint work with Tomoyuki Shirai
    (https://arxiv.org/abs/1903.04945).

    File: JpGerTimeTable0819.pdf

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  • Determinantal point processes in two-dimensional and higher-dimensional spaces Invited International conference

    KATORI Makoto

    Interactions between commutative and non-commutative probability, JSPS Program of France-Japan Bilateral Joint Seminars  ( Faculty of Science, Kyoto University )   2019.8  the JSPS Program of France-Japan Bilateral Joint Seminar

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    Language:English   Presentation type:Oral presentation (invited, special)  

    First I will explain a general framework of determinantal
    point processes (DPPs) based on the notions of
    partial isometries of Hilbert spaces
    and orthonormal functions.
    As examples of DPPs constructed by this framework,
    several kinds of DPPs are discussed in the two-dimensional
    spaces (e.g., a complex plane, a torus, a cylinder, and a sphere)
    as well as in higher-dimensional spaces.
    Universal properties observed in the bulk scaling limit
    are shown. Systems of stochastic differential equations
    having these DPPs as invariant measures are discussed.
    The present talk is based on the paper
    to be published in Commun. Math. Phys.
    (https://doi.org/10.1007/s00220-019-03351-5)
    and on a joint work with Tomoyuki Shirai
    (https://arxiv.org/abs/1903.04945).

    File: JFprogram.pdf

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  • Gaussian Free Fields with Boundary Points, Multiple SLEs, and Log-Gases Invited International conference

    KATORI Makoto

    The 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI) `Stochastic Analysis, Random Fields and Integrable Probability’  ( Kyushu University )   2019.8  Mathematical Society of Japan

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    A quantum surface (QS) (resp. an imaginary surface (IS)) is
    an equivalence class of pairs of simply commenced domains
    $D \subsetneq {\mathbb{C } }$ and the Gaussian free fields (GFFs) on $D$
    with the free (resp. Dirichlet) boundary condition
    induced by the conformal equivalence for random metric spaces.
    We define a QS with $N+1$ marked boundary points (MBPs)
    and an IS with $N+1$ boundary condition changing points (BCCPs)
    on $\partial D$ with $N \in {\mathbb{Z } }_{\geq 1}$, in which
    the real (resp. imaginary) part of the sum of ${\mathbb{C } }$-valued
    logarithmic (2D Coulomb) potentials arising from the MBPs (resp. BCCPs)
    is added to GFF in $D$.
    We consider the situation such that the boundary points evolve in time
    as a stochastic log-gas on $\partial D$ and
    multiple random slits are generated in $D$ by the
    multiple Schramm--Loewner evolution (SLE) driven by
    that stochastic log-gas.
    Then we cut the domain $D$ along the SLE slits, restrict the GFF
    on the resulting domain, and pull it back to $D$ following
    the reverse flow of the multiple SLE.
    We prove that if the log-gases on $\partial D$ follow
    the stochastic differential equations well-studied
    in random matrix theory ({\it e.g.}, Dyson's Brownian
    motion model, the Bru--Wishart processes),
    and parameters are properly chosen,
    then the coupled systems of GFFs and multiple SLE slits
    provide stationary processes.
    The obtained random systems are used to solve
    interesting geometric problems called
    the conformal welding problem and the flow line problem.
    The present study extends the previous
    results for a QS with two MBPs reported by Sheffield
    and for an IS with two BCCPs by Miller and Sheffield.
    This is a joint work with Shinji Koshida (Chuo University);
    see https://arxiv.org/abs/1903.09925.

    File: MSJ-SIprogram.pdf

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  • Quantum surface with marked boundary points and multiple SLE driven by Dyson model Invited International conference

    KATORI Makoto

    Japan Netherlands Workshop: Probabilistic Methods in Statistical Mechanics of Random Media  ( Lorentz Center, Leiden University )   2019.5  Mathematical Institute, Leiden University

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    File: Japan_Netherlands_May2019.pdf

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  • Partial Isometries, Duality, and Determinantal Point Processes Invited International conference

    KATORI Makoto

    Workshop on Random Matrices, Stochastic Geometry and Related Topics  ( National University of Singapore (NUS )   2019.3  National University of Singapore (NUS

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    A determinantal point process (DPP) is
    an ensemble of random $\mathbb{Z}_{\geq 0}$-valued
    Radon measures $\Xi$ on
    a space $S$ with measure $\lambda$,
    whose correlation functions are all
    given by determinants specified by an
    integral kernel $K$ called the correlation kernel.
    We consider a pair of Hilbert spaces,
    $H_{\ell}, \ell=1,2$, which are assumed to
    be realized as $L^2$-spaces,
    $L^2(S_{\ell}, \lambda_{\ell})$, $\ell=1,2$,
    and introduce
    a linear operator ${\cal W} : H_1 \mapsto H_2$
    and its adjoint ${\cal W}^{\ast} : H_2 \mapsto H_1$.
    We prove that if both of ${\cal W}$ and ${\cal W}^{\ast}$ are
    partial isometries and both of ${\cal W}^{\ast} {\cal W}$ and
    ${\cal W} {\cal W}^{\ast}$
    are of locally trace class, then we have unique pair
    of DPPs, $(\Xi_{\ell}, K_{\ell}, \lambda_{\ell})$, $\ell=1,2$,
    which satisfy duality relations.
    We assume that ${\cal W}$ admits an integral kernel $W$
    on $L^2(S_1, \lambda_1)$,
    and give practical setting of $W$ which makes
    ${\cal W}$ and ${\cal W}^{\ast}$ satisfy the above conditions.
    In order to demonstrate that the class of DPPs
    obtained by our method is large enough to
    study universal structures in a variety of DPPs,
    we show many examples of DPPs
    in one-, two-, and higher-dimensional spaces $S$,
    where several types of weak convergence from finite DPPs
    to infinite DPPs are given.
    One-parameter ($d \in \mathbb{Z}_{\geq 1}$) series of DPPs
    on $S=\mathbb{R}^d$ and $\mathbb{C}^d$ are discussed,
    which we call the Euclidean and the Heisenberg
    families of infinite DPPs, respectively, following
    the terminologies of Zelditch.
    This is a joint work with Tomoyuki Shirai.

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  • Universality of Determinantal Point Processes on Riemannian Manifolds Invited

    KATORI Makoto

    ( Shizuoka University, Department of Science )   2019.3  Shizuoka University, Department of Scence

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  • Determinantal point processes in high-dimensional spaces

    KATORI Makoto

    ( Nara Women's University, Department of Mathematics )   2019.1  Nara Women's University, Department of Mathematics

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  • Determinantal point processes on planes, tori, and spheres Invited International conference

    KATORI Makoto

    Spectra of Random Operators and Related Topics  ( Graduate School of Human and Environmental Studies, Kyoto University )   2019.1  Graduate School of Human and Environmental Studies, Kyoto University

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  • Elliptic DPP, one-component plasma, and GFF Invited International conference

    Makoto Katori

    RIMS Research Project `Gaussian Free Fields and Related Topics'  ( Research Institute for Mathematical Sciences (RIMS), Kyoto University )   2018.9  Research Institute for Mathematical Sciences (RIMS), Kyoto University

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    We introduce new families of determinantal point processes (DPPs)
    on a complex plane ${\mathbb{C } }$, which are classified into seven types
    following the irreducible reduced affine root systems,
    $R_N=A_N$, $B_N$, $B^{\vee}_N$, $C_N$, $C^{\vee}_N$, $BC_N$, $D_N$,
    $N \in {\mathbb{N } }$.
    Their multivariate probability densities are totally elliptic functions
    with periods $(L, iW)$, $0 < L, W < \infty$, $i=\sqrt{-1}$.
    The construction is based on the orthogonality relations
    with respect to the double integrals over the fundamental domain,
    $[0, L) \times [0, iW)$, which are proved in this paper
    for the $R_N$-theta functions introduced by Rosengren and Schlosser.
    In the scaling limit $N \to \infty, L \to \infty$
    with constant density
    $\rho=N/(LW)$ and constant $W$, we obtain four types of
    DPPs with an infinite number of
    points on ${\mathbb{C } }$, which have periodicity with period $i W$.
    In the further limit $W \to \infty$ with constant $\rho$,
    they are degenerated into three infinite-dimensional DPPs.
    One of them is uniform on ${\mathbb{C } }$ and equivalent
    with the Ginibre point process
    studied in random matrix theory, while
    other two systems are isotropic viewed from the origin,
    but non-uniform on ${\mathbb{C } }$.
    We show that the elliptic DPP
    of type $A_N$ is identified with the particle section,
    obtained by subtracting the background effect,
    of the two-dimensional exactly solvable model
    for one-component plasma studied by Forrester.
    Other two exactly solvable models of one-component plasma
    are constructed associated with the elliptic DPPs
    of types $C_N$ and $D_N$.
    Relationship to the Gaussian free field (GFF) on a torus
    is discussed for these three exactly solvable plasma models.
    (For more details, see {\sf arXiv:math-ph/1807.08287}.)

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  • Limit theorems for interacting Brownian motions I, II, III Invited International conference

    Makoto Katori

    Mini Workshop: Modern theory of Particle Systems  ( Faculty of Pure and Applied Mathematics )   2018.6  Faculty of Pure and Applied Mathematics

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    Language:English   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

    Dyson model is a stochastic particle system in one dimension R, in which repulsive force acts between any pair of particles with strength proportional to the inverse of distance. This multivariate stochastic process is realized as the system of one-dimensional Brownian motions conditioned never to collide with each other. We can show that this many-body system is exactly solvable and of determinantal in the sense that any spatio-temporal correlation function is expressed by determinant and is controlled by a single continuous function called the correlation kernel. In this lecture, we assume the special initial configuration such that all N particles are concentrated on the origin and we discuss the limit theorems in N ⟶ ∞. Wigner's semicircle law, which is extensively studied in random matrix theory and free probability, is demonstrated as the law of large numbers (LLN), which describes the density profile of particles in R at each time. Two kinds of limits called the bulk scaling limit and the soft-edge scaling limit are introduced in order to obtain determinantal processes with an infinite number of particles. As the central limit theorem (CLT) associated with the latter scaling limit, the Tracy--Widom distribution is discussed.

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  • Bessel Processes, Schramm-Loewner Evolution, and the Dyson Model Invited International conference

    Makoto Katori

    Universite D'Angers, Faculte des Sciences, Department de mathematiques, Colloquium  2018.6 

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  • Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal processes Invited International conference

    Makoto Katori

    Random Matrices and their Applications  ( kyoto University, Department of Mathematics )   2018.5  kyoto University, Department of Mathematics

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  • Orthogonal theta functions and elliptic determinantal point processes

    Workshop on "Random matrices, determinantal processes and their related topics" in Beppu 2018  2018 

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  • Orthogonal theta functions and elliptic determinantal point processes

    Workshop on "Random matrices, determinantal processes and their related topics" in Beppu 2018  2018 

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  • Hydrodynamic Limits of Multiple SLE Invited International conference

    Makoto Katori

    Tokyo-Seoul Conference in Mathematics - Probability Theory -  2017.12 

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  • Hydrodynamic limit of multipe SLE

    Makoto Katori

    2017.11 

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  • Excursion Processes Associated with Elliptic Combinatorics Invited International conference

    Makoto Katori

    The 16th International Symposium on Stochastic Analysis on Large Scale Interacting Systems  2017.11 

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  • q-拡張から楕円拡張へ Invited

    香取眞理

    第5回 Yokohama Workshop on Quantum Walks  2017.9 

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  • Elliptic Dyson models Invited International conference

    Makoto Katori

    Workshop on Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics  2017.3 

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  • Conformal invariance of complex Brownian motion and integrable stochastic particle systems

    Half-Day Workshop on Pattern Formation and Statistical Physics with Prof. Helmut R. Brand  2017 

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  • Relaxation Processes in the Non-Equilibrium Dyson Model with an Infinite Number of Particles Invited

    Kick off Meeting for Stochastic Anakysis on Infinite Particle Systems I  2017.1 

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  • Conformal invariance of complex Brownian motion and integrable stochastic particle systems

    Half-Day Workshop on Pattern Formation and Statistical Physics with Prof. Helmut R. Brand  2017 

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  • From q-probability to elliptic-probability

    5th Yokohama Workshop on Quantum Walks  2017 

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  • Hydrodynamic limit of multiple SLE

    2017 

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  • Dyson 模型の非平衡解に見られる緩和過程 Invited

    香取眞理

    ランダム作用素のスペクトルと関連する話題  2016.12 

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  • Martingales and determinantal processes Invited International conference

    Makoto Katori

    The 15th Workshop on Stochastic Analysis on Large Scale Interacging Systems  2016.11 

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  • Complex Nartingales and Determinantal Structures in Nonequilibrium Interacting Particle Systems International conference

    Makoto Katori

    26th IUPAP International Conference on Statistical Physics (StatPhys26)  2016.7 

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  • Complex martingales and determinantal processes Invited International conference

    Makoto Katori

    New approaches to non-equilibrium and random systems: KPZ integrability, universality, applications and experiments  2016.3 

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  • Complex martingales and determinantal structures in nonequilibrium interacting particle systems

    千葉大確率論研究会  2016 

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  • Determinantal interacting particle systems Invited International conference

    Makoto Katori

    International Symposium `RIKKYO MathPhys 2016'  2016.1 

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  • Complex martingales and determinantal structures in nonequilibrium interacting particle systems

    Chiba University Probability Theory Workshop  2016 

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  • Complex Martingales and Determinantal Structures in Nonequilibrium Interacting Particle Systems

    26th IUPAP International Conference on Statistical Physics (StatPhys26)  2016 

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  • Relaxation processes in the non-equilibrium Dyson model with an infinite number of particles

    Spectra of Randomn Operators and Related Topics  2016 

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  • Noncolliding pinned Brownian motions Invited International conference

    Makoto Katori

    Spectra of Random Operators and Related Topics  2015.12 

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  • ダイナミカルなランダム行列と棲み分けの問題 Invited

    香取眞理

    生命ダイナミクスの数理とその応用:理論からのさらなる深化  2015.12 

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  • Bessel Process, Schramm-Loewner Evolution, and the Dyson Model Invited International conference

    Makoto Katori

    RMT2015: Random matrix theory from fundamental mathematics to biological applicatoins  2015.11 

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  • Elliptic Bessel Process and Elliptic Dyson Model Realized as Temporally Inhomogeneous Processes Invited International conference

    Makoto Katori

    14th Stochastic Analysis on Large Scale Interacting Systems  2015.10 

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  • 可換砂山模型の数理 Invited

    香取眞理

    日本物理学会2015年秋季大会  2015.9 

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  • Abelian sandpile models in statistical mechanics Invited International conference

    Makoto Katori

    Workshop on `Probabilistic models with determinantal structure'  2015.4 

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  • Noncolliding Brownian bridges associated with elliptic functions

    関西確率論セミナー  2015 

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  • Determinantal martingales and determinantal processes

    立教大学数理物理学研究センター 2015年度第3回定例セミナー  2015 

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  • Elliptic determinantal evaluations and diffusion processes Invited International conference

    Makoto Katori

    Spectra of Random Operators and Related Topics  2015.1 

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  • Noncolliding Brownian bridges associated with elliptic functions

    2015 

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  • Determinantal martingales and determinantal processes

    Seminar at Research Center for Mathematical Physics, Rikkyo University  2015 

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  • Martingales for determinantal log-gases Invited International conference

    Makoto Katori

    YITP workshop `Interface fluctuations and KPZ universality class - unifying mathematical, theoretical, and experimental approaches'  2014.12 

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  • Elliptic determinantal processes Invited International conference

    Makoto Katori

    The 13th Workshop on Stochastic Analysis on Large Scale Interacting Systems  2014.11 

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  • Determinantal martingales and determinantal processes International conference

    Makoto Katori

    International Conference on Stochastic Processes, Analysis and Mathematical Physics  2014.8 

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  • 行列式過程としてダイソン模型をどれだけ拡張できるか

    研究集会「無限粒子系と確率場の諸問題IX」  2014 

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  • 時間発展するランダム行列

    第21回沼津研究会  2014 

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  • 時間発展するランダム行列固有値模型の揺動と相関

    2014年度夏学期第10回駒場物性セミナー  2014 

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  • Extensions of Dyson's model keeping determinantal solvability

    2014 

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  • Complex Brownian motion representation and Eynard-Mehta-type correlation kernel

    Workshop on ``Determinantal Processes and Related Topics"  2013 

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  • Complex Brownian motion representation of the Dyson model

    関西確率論セミナー 拡大セミナー  2013 

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  • Conformal martingale representations of log-gases [PP2-3-17 Poster]

    25th IUPAP International Conference on Statistical Physics (STATPHYS25)  2013 

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  • Determinantal martingales and determinantal processes

    Workshop `Random analytic functions, random matrices, and determinantal processes'  2013 

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  • Determinantal martingales and correlations of noncolliding random walks

    12th workshop on Stochastic Analysis on Large Scale Interacting Systems  2013 

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  • An elliptic extension of Dyson's Brownian motion model

    Workshop `New Topics on Stochastic and Quantum Interacting Particle Systems'  2013 

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  • Determinantal martingales and noncolliding diffusion processes

    Workshop `Spectra of Random Operators and Related Topics'  2013 

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  • Complex Brownian motion representation and Eynard-Mehta-type correlation kernel

    Workshop on ``Determinantal Processes and Related Topics"  2013 

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  • Complex Brownian motion representation of the Dyson model

    Kansai Probability Seminar (Kyoto Seminar) Extended  2013 

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  • Conformal martingale representations of log-gases

    25th IUPAP International Conference on Statistical Physics (STATPHYS25)  2013 

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  • Determinantal martingales and determinantal processes

    Workshop `Random analytic functions, random matrices, and determinantal processes'  2013 

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  • Determinantal martingales and correlations of noncolliding random walks

    12th workshop on Stochastic Analysis on Large Scale Interacting Systems  2013 

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  • An elliptic extension of Dyson's Brownian motion model

    Workshop `New Topics on Stochastic and Quantum Interacting Particle Systems'  2013 

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  • Determinantal martingales and noncolliding diffusion processes

    Workshop `Spectra of Random Operators and Related Topics'  2013 

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  • Vicious Brownian motion, O'Connell's process, and equilibrium Toda lattice Invited International conference

    Makoto Katori

    EPSRC Symposium Workshop on Interacting particle systems, growth models, and random matrices  2012.3 

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  • 条件付ブラウン運動・SLE・ランダム行列・量子戸田格子

    横浜国立大工学部 玉野セミナー  2012 

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  • 条件付ブラウン運動・SLE・ランダム行列・量子戸田格子

    行列模型とその周辺  2012 

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  • Vicious Brownian motion, O'Connell process, and equilibrium Toda lattice

    Workshop on Recent Topics of Statistical Mechanics and Probability Theory  2012 

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  • System of complex Brownian motions associated with the O'Connell process

    研究集会「無限粒子系、確率場の諸問題 VIII」  2012 

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  • Determinantal formula appearing in non-determinantal process

    Spectra of Random Operators and Related Topics  2012 

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  • Vicious Brownian motions, entire functions, and quantum Toda lattice

    Novel Development of Statistical Physics  2012 

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  • Vicious Brownian motion, O'Connell process, and equilibrium Toda lattice

    Workshop on Recent Topics of Statistical Mechanics and Probability Theory  2012 

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  • System of complex Brownian motions associated with the O'Connell process

    2012 

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  • Determinantal formula appearing in non-determinantal process

    Spectra of Random Operators and Related Topics  2012 

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  • Vicious Brownian motions, entire functions, and quantum Toda lattice

    Novel Development of Statistical Physics  2012 

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  • On Matsumoto-Yor process and O'Connell process

    研究集会「無限粒子系、確率場の諸問題 VII」  2011 

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  • Vicious Brownian motion with Toda lattice potential and O'Connell's process

    ランダム作用素のスペクトルと関連する話題  2011 

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  • On Matsumoto-Yor process and O'Connell process

    2011 

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  • Vicious Brownian motion with Toda lattice potential and O'Connell's process

    2011 

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  • Determinantal Processes and Entire Functions International conference

    Makoto Katori

    34th Conference on Stochastic Processes and Their Applications  2010.9 

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  • 「統計力学模型とSLE」「SLEに関する話題」

    勉強会「Loewner 方程式とSLE」  2010 

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  • Stochastic Loewner Evolutions and Statistical Mechanics

    第 53 回函数論シンポジウム  2010 

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  • 非衝突ブラウン運動・ランダム行列・整関数

    第9回岡シンポジウム  2010 

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  • Dyson 模型の複素ブラウン運動表示と行列式型相関関数

    大岡山談話会  2010 

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  • Stochastic Loewner Evolutions and Statistical Mechanics

    2010 

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  • Entire functions and relaxation processes of infinite particle systems Invited International conference

    Makoto Katori

    Conference on Probability and Stochastic Processes  2009.11 

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  • Zeros of Airy Function and Relaxation Process Invited International conference

    Makoto Katori

    International Conference `Selfsimilar Processes and their Applications'  2009.7 

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  • 連続関数空間上の共形不変な確率測度とSchramm-Loewner Evolution Invited

    香取眞理

    日本数学会2009年度年会  2009.3 

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  • Non-Equilibrium Dynamics of Dyson's Model with Infinite Particles

    古典および量子ダイナミクス・非平衡統計力学に関するワークショップ  2009 

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  • SLE/CFT correspondence

    Workshop on "SLE and related topics"  2009 

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  • ディラック方程式・量子ウォーク・グラフェン

    量子ウォークの相対論的記述と固有値解析  2009 

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  • 臨界現象・フラクタル曲線とSchramm-Loewner Evolution

    Summer School 数理物理 2009  2009 

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  • 数理物理学の新展開--ランダム行列とSLE

    第54回物性若手夏の学校  2009 

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  • Zeros of entire functions and relaxation processes

    VIIIth Symposium quot;Stochastic Analysis on Large Scale Interacting Systems"  2009 

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  • Dyson's Brownian motion model with beta=2 and entire functions

    ランダム作用素のスペクトラルと関連する話題  2009 

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  • SLE/CFT correspondence

    Workshop on "SLE and related topics"  2009 

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  • Zeros of entire functions and relaxation processes

    VIIIth Symposium "Stochastic Analysis on Large Scale Interacting Systems"  2009 

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  • Dyson's Brownian motion model with beta=2 and entire functions

    2009 

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  • Multiple orthogonal polynomials and noncolliding diffusion processes Invited

    Makoto Katori

    2008.11 

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  • Multiple orthogonal polynomials and noncolliding Brownian motions Invited International conference

    Makoto Katori

    Workshop on Stochastic Analysis on Large Scale Interacting Systems  2008.11 

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  • 確率微分方程式と相転移・臨界現象 Invited

    香取眞理

    第13回久保記念シンポジウム,井上科学振興財団  2008.10 

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  • 非衝突ブラウン運動・多重エルミート関数・ランダム行列 Invited

    香取眞理

    九州大学大学院理学研究院物理部門談話会,九州大学大学院理学府  2008.7 

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    Language:Japanese   Presentation type:Public lecture, seminar, tutorial, course, or other speech  

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  • Noncolliding Brownian motions with arbitrary initial configuration and multiple Hermite polynomials Invited International conference

    Makoto Katori

    Workshop on Combinatorics and Statistical Physics/Erwin Schrodinger International Institute for Mathematical Physics (ESI)  2008.5 

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  • Maximum height distribution of noncolliding Bessel bridges Invited International conference

    Makoto Katori

    ESI Programme on Combinatorics and Statistical Physics/Erwin Schrodinger International Institute for Mathematical Physics (ESI)  2008.3 

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  • Multiple orthogonal polynomials and determinantal processes Invited International conference

    Makoto Katori

    Workshop on Random matrices, special functions and related topics/Research Institute for Mathematical Science (RIMS), Kyoto University  2008 

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  • An Introduction to Schramm-Loewner Evolution

    研究集会「統計物理に関連する数学的話題」,東工大理工学研究科数学教室  2007 

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  • Schramm-Loewner Evolution (SLE) and Conformal Restriction

    保型形式・無限可積分系合同合宿(2007),東京大学大学院数理科学研究科  2007 

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  • ブラウン運動・SLE・2次元臨界現象(Lawler, Schramm, Werner らの結果の紹介)

    非可換解析とミクロ・マクロ双対性,京都大学数理解析研究所  2007 

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  • Noncolliding diffusion processes and determinantal processes

    ランダム作用素のスペクトルと関連する話題,京都大学大学院人間・環境学研究科  2007 

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  • ブラウン運動とリーマンのゼータ関数

    立教大学理論物理学研究室セミナー  2007 

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  • Noncolliding diffusion processes and determinantal processes

    2007 

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  • 量子ウォークの特異な拡散現象

    九州大学大学院工学研究院特別講義,九州大学大学院工学研究院エネルギー量子工学部門  2006 

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  • Wigner formula of rotation matrices and quantum walks

    研究集会「非線型波動現象における基礎理論,数値計算および実験のクロスオーバー」,九州大学応用力学研究所  2006 

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  • 量子ウォークの特異な極限分布

    立教大学理論物理学研究室セミナー  2006 

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  • 時間的に非斉次な非衝突ブラウン運動

    東京無限可積分セミナー,東京大学数理科学研究科  2005 

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  • ワイル粒子の軌道状態と量子ウォークの極限分布

    日本物理学会 2005 年度秋季大会,同志社大学,日本物理学会  2005 

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  • 量子ウォークとワイル方程式

    日本物理学会 2005 年度秋季大会,同志社大学,日本物理学会  2005 

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  • 非衝突ブラウン運動と Tracy-Widom 分布

    研究集会「無限粒子系、確率場の諸問題」,奈良女子大学理学部  2005 

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  • 非衝突拡散粒子系とランダム行列理論

    日本物理学会第59回年次大会(九大),日本物理学会  2004 

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  • Vicious Walks and Random Matrices

    Satellite Meeting of STATPHYS 22 in Seoul, Korea,Nonequilibrium Statistical Physics of Complex Systems/Korea Institute for Advanced Study (KIAS),IUPAP  2004 

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  • 非衝突拡散過程と四元数行列式

    可積分系数理の展望と応用,京都大学数理解析研究所  2004 

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  • Vicious Walks, Schur Functions, and Random Matrix Theory

    Joint Seminar in Probability (Chinese Academy of Science, Peking Univ., Beijing Normal Univ., China)/Inst. Applied Math., Chinese Academy of Science, China  2004 

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  • Matrix-valued Stochastic Processes and Noncolliding Diffusion Particle Systems

    Workshop on Infinite Particle Systems and Critical Phenomena/Beijing Jiaotong University  2004 

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  • Nonintersecting paths, noncolliding diffusion processes and representation theory

    表現論における組み合わせ論的手法とその応用/京都大学数理解析研究所  2004 

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  • 時間的に非斉次な非衝突ブラウン運動

    立教SFR研究会「弦理論と重力理論の数学的構造解明に関する学際的研究」,立教大学学術推進特別重点資金(SFR)自由プロジェクト研究  2004 

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  • Vicious Walks and Random Matrices

    Satellite Meeting of STATPHYS 22 in Seoul, Korea,Nonequilibrium Statistical Physics of Complex Systems/Korea Institute for Advanced Study (KIAS),IUPAP  2004 

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  • Vicious Walks, Schur Functions, and Random Matrix Theory

    Joint Seminar in Probability (Chinese Academy of Science, Peking Univ., Beijing Normal Univ., China)/Inst. Applied Math., Chinese Academy of Science, China  2004 

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  • Matrix-valued Stochastic Processes and Noncolliding Diffusion Particle Systems

    Workshop on Infinite Particle Systems and Critical Phenomena/Beijing Jiaotong University  2004 

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  • Nonintersecting paths, noncolliding diffusion processes and representation theory

    Combinatorical Methods in Representation Theory/RIMS, Kyoto University  2004 

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  • 無限粒子 vicious walk とエアリー過程

    日本物理学会  2003 

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  • Noncolliding systems of diffusion particles and multi-matrix models

    大規模相互作用系の確率解析(日本数学会シンポジウム)  2003 

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  • Noncolliding Brownian particles, Schur polynomials, and GUE/GOE two-matrix model

    京都大学基礎物理学研究所セミナー,京都大学基礎物理学研究所  2003 

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  • 非衝突ランダムウォーク, シューア関数, 多層行列模型

    非線形波動および非線形力学系の数理とその応用(九大応用力学研究所研究集会;研究集会報告 2004年4月),九州大学応用力学研究所  2003 

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  • 非衝突拡散粒子系とランダム行列理論

    確率モデルの統計力学(京大基研研究集会),京都大学基礎物理学研究所  2003 

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  • Noncolliding Brownian particles, Schur polynomials, and GUE/GOE two-matrix model

    2003 

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  • 繰り込み群による vicious walk の研究

    日本物理学会  2002 

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  • vicious walks の連続極限とランダム行列

    日本物理学会  2002 

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  • Scaling limit of vicious walkers, Schur functions and Gaussian random matrix ensembles

    Oxford University, UK  2001 

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  • Scaling limit of vicious walkers, Schur functions and Gaussian random matrix ensembles

    Oxford University, UK  2001 

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  • SOC as a critical phenomena in a nonconservative Abelian sandpile model

    ICTP,Trieste,Italy  2000 

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  • SOC as a critical phenomena in a nonconservative Abelian sandpile model

    ICTP,Trieste,Italy  2000 

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  • 粒子消滅を伴う可換砂山モデルの非臨界性

    日本物理学会第54回年会,広島大学,日本物理学会  1999 

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  • 3次元Directed Percolationの臨界挙動

    日本物理学会第54回年会,広島大学,日本物理学会  1999 

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  • バッタの相変異における協同現象

    日本物理学会第54回年会,広島大学,日本物理学会  1999 

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  • Chiral Potts Models, Friendly Walkers and Directed Percolation Problem;「Markov 連鎖とその統計力学への応用」

    平成9年度科研費基盤研究(A)「確率論の総合的研究」研究会,神戸大(招待講演)  1998 

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  • 非等方的な可換砂山モデルの厳密解

    日本物理学会第53回年会,東邦大学,日本大学  1998 

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  • Forest dynamics with canopy gap expansion and stochastic Ising model

    STATPHYS 20, Paris, France/IUPAP  1998 

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  • Effect of anisotropy on the self-organized critical states of Abelian sandpile models

    International Conference on Percolation and Disordered Systems,Giessen, Germany/International Conference on Percolation and Disordered Systems,Giessen,Germany.  1998 

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  • Chiral Potts models, friendly walkers and directed percolation problem

    STATPHYS 20, Paris, France IUPAP  1998 

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  • A stochastic lattice model for locust outbreak

    International Conference on Percolation and Disorderd Systems, Giessen, Germany/International Conference on Percolation and Disordered Systems,Giessen,Germany.  1998 

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  • 砂山モデルにおけるなだれサイズ分布の多様性

    日本物理学会秋の分科会,琉球大学,沖縄国際大学,日本物理学会  1998 

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  • 非等方的な可換砂山モデルの厳密解Ⅱ

    日本物理学会秋の分科会,琉球大学,沖縄国際大学,日本物理学会  1998 

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  • 森林の林冠ギャップの空間分布とイジング模型Ⅱ

    日本物理学会秋の分科会,琉球大学,沖縄国際大学,日本物理学  1998 

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  • Forest dynamics with canopy gap expansion and stochastic Ising model

    STATPHYS 20, Paris, France/IUPAP  1998 

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  • Effect of anisotropy on the self-organized critical states of Abelian sandpile models

    International Conference on Percolation and Disordered Systems,Giessen, Germany/International Conference on Percolation and Disordered Systems,Giessen,Germany.  1998 

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  • Chiral Potts models, friendly walkers and directed percolation problem

    STATPHYS 20, Paris, France IUPAP  1998 

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  • A stochastic lattice model for locust outbreak

    International Conference on Percolation and Disorderd Systems, Giessen, Germany/International Conference on Percolation and Disordered Systems,Giessen,Germany.  1998 

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  • クラスター変分近似の収束性とコヒーレント異常

    クラスター変分法に関するミニシンポジウム,東工大(招待講演)  1997 

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  • Chiral Potts模型とDirected Percolation II

    日本物理学会年会,名城大  1997 

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  • Chiral Potts模型とDirected Percolation I

    日本物理学会講演概要集、日本物理学会年会,名城大  1997 

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  • 相互作用する randem walks の数え上げと Chiral Potts 模型

    日本物理学会秋の分科会,神戸大  1997 

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  • 森林の林冠ギャップの空間分布とイジング模型

    日本物理学会講演概要集、日本物理学会秋の分科会,神戸大  1997 

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  • 一般化した砂山モデルについて

    日本物理学会秋の分科会,神戸大  1997 

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  • Cluster Variational Approach to Non-equilibrium Lattice Models Invited

    M.Katori

    Theory and Applications of the Cluster Variation and Path Probability Methods ed J.L.Moran-Lopez and J.M.Sanchez  1996.4 

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  • Directed Percolationの展開係数における規則性II

    日本物理学会(第51回年会)講演概要集,日本物理学会  1996 

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  • 非対称Directed Percolationの級数展開

    日本物理学会秋の分科会,山口大  1996 

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  • 新しい砂山モデルに対する平均場近似とモンテカルロシミュレーション

    日本物理学会秋の分科会,山口大  1996 

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  • Directed Percolationの級数展開係数のballot数表示

    日本物理学会講演概要集、日本物理学会秋の分科会,山口大  1996 

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  • 自己組織化臨界現象を示す幾つかのモデル

    「パーコレーション,無限粒子系の数理」(平成8年度科学研究費補助金 基盤研究A(1))「確率論の総合的研究」,横浜国立大(招待講演)  1996 

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  • Catalan numbers in the percolation probability series for the directed square lattice

    「パーコレーション,無限粒子系の数理」(平成8年度科学研究費補助金 基盤研究A(1))「確率論の総合的研究」,横浜国立大(招待講演)  1996 

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  • Catalan numbers in the percolation probability series for the directed square lattice

    1996 

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  • 無限粒子系の非可逆な確率過程の研究

    中央大学理工学研究所年報,中央大学理工学研究所  1995 

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  • 非平衡格子モデルにおけるDualityとUniversality

    日本物理学会(第50回年会)講演概要集,日本物理学会  1995 

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  • Contact Process, Directed Percolation and Beyond

    Meeting on Pattern Formation (KEK),高エネルギー研究所  1995 

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  • Cluster variational Approach to Non-equilibrium Lattice Models

    International Workshop on the Theory and Applications of the Cluster Variation and Path Probability Methods, (Mexico) Proceedings/Plenum Press  1995 

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  • Time-Reversal Duality and Planar Lattice Duality in Non-equilibrium Lattice Models

    International Conference on Future of Fractals(Aichi),/World Scientific Pub. Co.  1995 

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  • クラスター近似による砂山モデルの高さ分布の計算

    日本物理学会(1995年秋の分科会)講演概要集,日本物理学会  1995 

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  • 確率的セルラ・オートマンのedgeプロセスとその裏格子プロセス

    Fractals日本物理学会(1995年秋の分科会)講演概要集,日本物理学会  1995 

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  • Stochastic Cellular Automataに関する最近の話題

    平成7年度科研費報告書「確率論の総合的研究」,平成7年度科研費基盤研究A(1)  1995 

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  • Contact Process, Directed Percolation and Beyond

    Meeting on Pattern Formation (KEK)  1995 

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  • Cluster variational Approach to Non-equilibrium Lattice Models

    International Workshop on the Theory and Applications of the Cluster Variation and Path Probability Methods, (Mexico) Proceedings/Plenum Press  1995 

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  • Time-Reversal Duality and Planar Lattice Duality in Non-equilibrium Lattice Models

    International Conference on Future of Fractals(Aichi),/World Scientific Pub. Co.  1995 

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  • Introduction to Non-equilibrium Lattice Models

    東北大学金属材料研究所セミナー  1994 

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  • 高次元における拡散を伴ったコンタクト・プロセスの相図

    日本物理学会(第49回年会)講演概要集,日本物理学会  1994 

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  • The Holley-Liggett argument revisited : an application to the generalized contact process

    ICM94 Abstracts/International Mathematical Union  1994 

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  • 1次元非対称exclusion processの定常分布について

    日本物理学会(1994年秋の分科会)講演概要集,日本物理学会  1994 

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  • Introduction to Non-equilibrium Lattice Models

    1994 

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  • The Holley-Liggett argument revisited : an application to the generalized contact process

    ICM94 Abstracts/International Mathematical Union  1994 

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  • Phase transition in contact process and its related processes

    確率論セミナー主催シンポジウム講演概要集,確率論セミナー  1993 

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  • 粒子系における非可逆な定常状態について

    数理解析研講究録,京大数理解析研究所  1993 

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  • 拡張された1次元コンタクト・プロセスの臨界値に対する評価II

    日本物理学会(1993年秋の分科会)講演概要集,日本物理学会  1993 

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  • 1次元Pain Annihilation Model における相転移の存在証明

    日本物理学会(1993年秋の分科会)講演概要集,日本物理学会  1993 

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  • 伝染のモデルと交通渋滞のモデル

    東北大学理学部物理セミナー  1993 

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  • コンタクト・プロセスの相転移現象

    東北大学工学部数物談話会  1993 

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  • Phase transition in contact process and its related processes

    1993 

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  • 触媒表面の数理モデル

    物性研究,物性研究刊行会  1992 

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  • 無限粒子系の物理と数学

    物性研究,物性研究刊行会  1992 

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  • 拡張された1次元コンタクト・プロセスの臨界値に対する評価

    日本物理学会講演概要集,日本物理学会  1992 

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  • 拡散を伴うコンタクト・プロセスの相図について

    物性研だより,東京大学物性研究所  1992 

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  • 拡散を伴ったコンタクト・プロセスについて

    物性研究,物性研究刊行会  1992 

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  • 非可逆な化学反応モデルにおける拡散の効果について

    物性研究,物性研究刊行会  1991 

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  • コンタクト・プロセスにおける感染領域の定常分布について

    物性研究,物性研究刊行会  1990 

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  • Contact Processの定常状態における相転移現象について

    統計数理,統計数理研究所  1990 

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  • Window-CID-CAMによるContact Processの研究

    物性研究,物性研究刊行会  1989 

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  • カノニカル・シリーズの作り方-クラスター平均場近似と相関等式切断近似-

    物性研究,物性研究刊行会  1989 

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  • 相転移とCAM理論

    物性研究,物性研究刊行会  1988 

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  • GT線における非線形帯磁率について

    物性研究,物性研究刊行会  1986 

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  • レプリカ理論と非線形帯磁率

    物性研究,物性研究刊行会  1985 

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Awards

  • 中央大学学術研究奨励賞

    2022.2   Chuo University  

    Makoto Katori

  • The MSJ Analysis Prize

    2021.9   The Mathematical Society of Japan   Interacting particle systems related to statistical mechanics

    Makoto Katori

  • Editor's Pick

    2020.8   American Institute of Physics   Conformal welding problem, flow line problem, and multiple Schramm-Loewner evolution

    Makoto Katori, Shinji Koshida

  • JPSJ Papers of Editors' Choice

    2015.10   Journal of the Physical Society of Japan, The Physical Society of Japan   Self-Elongation with Sequential Folding of a Filament of Bacterial Cells

    Ryojiro Honda, Jun-ichi Wakita, Makoto Katori

  • 第5回 井上科学研究奨励賞

    1989  

  • 5th Inoue Research Award for Young Scientists

    1989  

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Research Projects

  • Development, evolution, and new development of stochastic analysis of infinite particle systems

    Grant number:21H04432  2021.4 - 2026.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (A) 

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    Grant amount: \41340000 ( Direct Cost: \31800000 、 Indirect Cost: \9540000 )

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  • フェルミオン点過程と共形不変SLE曲線による確率場の総合的理論の構築

    2019.4 - 2024.3

    日本学術振興会  科学研究費補助金、基盤研究(C) 

    香取 眞理

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    Authorship:Principal investigator  Grant type:Competitive

    Grant amount: \4290000 ( Direct Cost: \3300000 、 Indirect Cost: \990000 )

    ランダム行列理論は1次元線上や2次元平面上のランダムな点集合であるフェルミオン点過程の確率理論を与え,平面上の臨界統計力学模型とフラクタル模型のスケーリング連続極限を表現するシュラム・レヴナー発展(SLE)は共形変換不変な確率測度をもつランダムな曲線の時間発展理論である。本研究ではこれら点や曲線に対する確率理論の研究を深め,それらで特徴づけられる確率場に対して総合的な新理論を構築する。具体的には,高次元リーマン多様体上のフェルミオン点過程の一般論を展開し,1成分あるいは2成分プラズマ模型を経由して自由ガウス場,さらにそのリウビル量子重力場への変換を明らかにする。局所的な場の接続を分類し,多重SLEでコントロールする。多重SLEの普遍的な確率法則の導出やDLA(拡散律速凝集)模型のフラクタル次元決定など,多くの未解決問題に挑戦する。

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  • Stochastic Analysis of Random Phenomena from the Viewpoint of Determinantal Point Processes

    2018.4 - 2022.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

    Shirai, Tomoyuki

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  • Stochastic Analysis for Infinite Particle Systems

    2016.4 - 2021.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

    Osada, Hirofumi

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    Grant type:Competitive

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  • New Theory of Fluctuations and Correlations in Exactly Solvable Models of Nonequilibrium Statistical Mechanics

    2014.4 - 2019.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research, Kiban (C) 

    Katori, Makoto

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  • Collective motion of bacterial cells in a two-dimensional circular pool

    Grant number:15K13537  2015.4 - 2018.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Challenging Exploratory Research  Chuo University

    Wakita Jun-ichi, KATORI Makoto, MATSUSHITA Mitsugu, YAMAMOTO Ken, NARIZUKA Takuma, HONDA Ryojiro, TSUKAMOTO Shota, HARADA Shohei, UMEDA Sora, NISHIOKA Mizuho

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    Grant amount: \2730000 ( Direct Cost: \2100000 、 Indirect Cost: \630000 )

    Recently, various types of collective behavior observed in swarming micro-organisms, marching insects, flocking birds, and walking pedestrians have been studied in the field of physics. The relation between individual behavior and collective behavior has been investigated from the view point of statistical physics. In our studies, we found that Bacillus subtilis cells as a kind of micro-organisms showed six types of collective behavior depending on the reduced cell length (defined as the ratio of cell length to pool diameter) and the cell density in a circular pool which was made on an agar plate surface. Then, we have confirmed that the interaction of a cell with the brim of a pool, the interaction between cells, and the fluctuation in the moving direction of cells are essential to the collective behavior.

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  • Study of Determinantal Point Processes and their Generalizations

    2014.4 - 2018.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research 

    Shirai, Tomoyuki

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  • 無限粒子系とランダム行列の確率解析

    Grant number:16H02149  2016.4 - 2017.3

    日本学術振興会  科学研究費助成事業  基盤研究(A)  九州大学

    長田 博文, 種村 秀紀, 白井 朋之, 香取 眞理, 熊谷 隆, 舟木 直久

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    Grant amount: \42640000 ( Direct Cost: \32800000 、 Indirect Cost: \9840000 )

    干渉ブラウン運動のtagged粒子の不変原理に関して、従来より適切かつ精密な定式化を行い、Kipnis-Varadhanの不変原理の対応物が成立することを証明した。
    従来は、Palm測度に対して、無限個の環境粒子系にたいして、tagged粒子のブラウン運動への収束が「測度収束」の意味で成立するという形で、主張が定式化されてきた。これは、80年代前半に行われたGuo-Papanicolauの研究以来の伝統的な定式化ではあるが、tagged粒子から見た他の無限個の粒子家の挙動を記述する確率微分方程式に対する解析となるため、すぐには元来の問題との関係が分かりにくい。
    それに対して、今回は、粒子のラベルを考慮し構造に入れることにより、元々の平行移動不変な平衡分布に関して、可逆な確率力学を記述する確率微分方程式を考え、その個々の粒子が、ブラウン運動に初期条件に関して測度収束するという、より自然なわかりやすい定式化となった。
    また、応用として、1次元の無限粒子系が互いに衝突しない(順序を入れ替えない)という性質をもつとき、極限が常に退化するという結果を得た。この事実自体は、当然の結果だが、ポイントはこれが常に成立するということを、幾何的な結果から平易に示したという一般性を備えている点である。
    ランダム行列に関係する干渉ブラウン運動は、対数関数(2次元クーロンポテンシャル)で相互作用する確率力学である。また、干渉ブラウン運動を含む広い範囲の無限次元確率微分方程式に対して一般論を構築し、更に発展させている最中だが、これに関する最近の研究結果をまとめてreview論文として報告した。

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  • Developing micro-macro-structure connection engineering for novel functions of Si-LED based on dressed photons

    Grant number:15K13374  2015.4 - 2016.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Challenging Exploratory Research  The University of Tokyo

    OHTSU Motoichi, KATORI Makoto, NARUSE Makoto

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    Grant amount: \3770000 ( Direct Cost: \2900000 、 Indirect Cost: \870000 )

    Highly efficient and high power Si-LEDs were fabricated by dressed-photon-phonon (DPP)-assisted annealing. It was found that Boron atoms, a DPP-creation source, formed a pair, and the separation between the two atoms in the pair was three-times the lattice constant of the Si crystal. The atom pair aligned along the plane normal to the direction of the incident light beam used for the annealing. It was also normal to the polarization direction of this light. Simultaneously with these experimental works, novel mathematical scientific model was developed to analyze the light-matter interaction in a mesoscopic space. The results of the analyses agreed well with the experimental results. Furthermore, statistical model of energy transfer in a random medium via dressed photons was developed.

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  • Stochastic geometry and dynamics of infinite particle systems interacting with two-dimensional Coulomb potential

    Grant number:24244010  2012.4 - 2016.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (A)  Kyushu University

    Osada Hirofumi, KOTANI Shinichi, KATORI Makoto, SHINPDA Masato, OTOBE Yoshiki

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    Grant amount: \28470000 ( Direct Cost: \21900000 、 Indirect Cost: \6570000 )

    We establish a general theory for solving infinite-dimensional stochastic differential equations (ISDE) with symmetry typically appearing in statistical mechanics. In particular, we prove the pathwise uniqueness and the existence of the strong solution under a very general framework. This method is novel, and regards the tail sigma field of the configuration space as a boundary of the ISDE. Furthermore, if the tail sigma field is trivial, then a strong solution exists. If the set of probability-one events is unique, then the pathwise uniqueness of solution holds.
    The method is effective for the ISDE with logarithmic interaction potentials, which appear in random matrix theory.

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  • Stochastic analysis of infinite particle systems in nonequilibruim

    Grant number:23540122  2011.4 - 2015.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Chiba University

    TANEMURA Hideki, KATORI Makoto, SASAMOTO Tomohiro

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    Grant amount: \4810000 ( Direct Cost: \3700000 、 Indirect Cost: \1110000 )

    We study existence and uniqueness of infinite dimensional stochastic differential equations (ISDEs). We clarified the relation between the uniqueness of solutions and the triviality of tail sigma fields. Applying the result, we obtained the uniqueness of strong solutions of ISDEs describing the systems of Brownian motions with long range interaction including logarithmic potential, and determined the Dirichlet forms associated with the systems.

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  • Study on stochastic processes with determinantal structure

    Grant number:22340020  2010.4 - 2014.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Kyushu University

    SHIRAI Tomoyuki, TANEMURA Hideki, KATORI Makoto, OSADA Hirofumi

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    Grant amount: \13780000 ( Direct Cost: \10600000 、 Indirect Cost: \3180000 )

    Correlation functions of the eigenvalues of certain random matrices can be expressed by using determinant. A point configuration whose correlation functions are determinants has repulsive property and is called a determinantal point process. Many mathematical objects can be described as a determinantal point process and analyzed precisely. We study the Ginibre point process, which is a typical model of determinantal point process, and we use this determinantal point process instead of poisson point process as a model of base stations of a cellular network and analyze it. We investigated determinantal point processes from both theoretical and applied mathematical points of view.

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  • Statistical Mechanics for Random Curves and Patterns with Applications

    2009.4 - 2014.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research, Kiban (C) 

    Katori, Makoto

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  • Infinite-dimensional stochastic dynamical systems motivated by random matrices and statistical physics

    Grant number:21340031  2009 - 2011

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Kyushu University

    OSADA Hirofumi, FUNAKI Tadahisa, TANEMURA Hideki, SHIRAI Tomoyuki, KATORI Makoto, OTOBE Yoshiki, SHINODA Masato, YANO Yuko, YANO Kouji

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    Grant amount: \16640000 ( Direct Cost: \12800000 、 Indirect Cost: \3840000 )

    We have established a general construction theorem and an SDE representation theorem for interacting Brownian motions with 2D Coulomb potentials. We have applied them to the representative random point fields arising from Random Matrix Theory such as Ginibre, Dyson, Bessel random point fields, and have detected and solve the infinite-dimensional stochastic differential equations describing the associated stochastic dynamics. We prove the Palm measures of Ginibre random point field have very strange property that is very different from usual Gibbs measures with Ruelle' s class interaction potentials. We have constructed the time evolutional model of 2D Young diagram and have proved its scaling limit.

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  • Particle systems with interaction

    Grant number:19540114  2007 - 2010

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Chiba University

    TANEMURA Hideki, KONNO Norio, KATORI Makoto, SASAMOTO Tomohiro

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    Grant amount: \4420000 ( Direct Cost: \3400000 、 Indirect Cost: \1020000 )

    A stochastic process is called determinantal if its correlation functions are represented by determinants. During the research we proved that for any fixed configuration the noncolliding Brownian motion and the noncolliding squared Bessel process are determinantal. When number of particle is infinite, we gave sufficient conditions so that the noncolliding processes are exist and have continuous paths. We also showed the relaxation phenomena for the processes.

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  • Random Matrix Theory for Condensed Matters

    2005.4 - 2009.3

    Japan Society for the Promotion of Science  Gtants-in-Aid for Scientific Research 

    Katori, Makoto

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  • Formation Mechanism of Growing Random Patterns

    Grant number:18340115  2006 - 2009

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Chuo University

    MATSUSHITA Mitsugu, KATORI Makoto, TAGUCHI Yoshihiro, MATSUYAMA Touhei, SUDA Junichiro, YAMAZAKI Yoshihiro

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    Grant amount: \11790000 ( Direct Cost: \10500000 、 Indirect Cost: \1290000 )

    We have mainly studied three topics on universal structural and statistical properties of random patterns seen in nature and our society. 1. We have established morphological diagrams on how colony patterns of bacterial species Escherichia coli and Pseudomonas aeruginosa change when environmental conditions vary. We have also studied the spatiotemporal change of cell density when periodic colony growth occurs. 2. We have investigated scaling properties, especially self-affinity, of growing interface of random patterns, by taking bacterial colonies and real landscapes as examples. 3. We have laid the foundation that the fundamental statistical aspects seen in complex systems in general are described by lognormal distributions, based on our observations of prefectural and municipal populations, our body height and weight, and so on.

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  • Interacting Infinite Particle Systems and Random Matrices

    Grant number:15540106  2003 - 2006

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  CHIBA UNIVERSITY

    TANEMURA Hideki, NAKAGAMI Jyunichi, NAGISA Masaru, KONNO Norio, KATORI Makoto

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    Grant amount: \3300000 ( Direct Cost: \3300000 )

    It is known that the distribution of particle positions in Dyson's model of Brownian motions coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brownian motions. We considered a system of noncolliding Brownian motions, in which the noncolliding condition is imposed in a finite time interval (0,T]. This is a temporally inhomogeneous diffusion process and in the limit T to infinity, it converges to Dyson's model. We constructed such a Hermitian matrix-valued process that its eigenvalues are identically distributed with the particle positions of our system of noncolliding Brownian motions.
    As an extension of the theory of Dyson's models for the standard Gaussian random-matrix ensembles, we made a systematic study of hermitian matrix-valued processes. In addition to the noncolliding Brownian motions, we introduced noncolliding systems of generalized meanders and showed that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems.
    Then we proved that these non-colliding diffusions are Pfaffian processes, in the sense that any multitime correlation function is given by a Pfaffian. In the infinite particle limit, we showed that the elements of matrix kernels of the obtained infinite Pfaffian processes are generally expressed by the Riemann-Liouville differintegrals of functions comprising the Bessel functions.

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  • Research on the formation and fluctuation of random shapes in mathematical models of statistical mechanics

    Grant number:12440027  2000 - 2002

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  KOBE UNIVERSITY

    HIGUCHI Yasunari, NAKANISHI Yasutaka, MIYAKAWA Tetsuro, FUKUYAMA Katsushi, MURAI Joushin, YAMAZAKI Tadashi

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    Grant amount: \10100000 ( Direct Cost: \10100000 )

    1. We found a new proof of the fact that there are only two extremal Gibbs states for the two dimensional Ising model based on the percolation argument. As an application of this new method, we proved that there are only two extremal points of translationally invariant Gibbs states for the two dimensional Widom-Rowlinson model for sufficiently low temperatures.
    2. We gave an estimate of the speed of convergence for the time constant of the first passage Ising percolation for temperatures above the critical point.
    3. We proved a Dobrushin-Hryniv type limit theorem for the two-dimensional Widom-Rowlinson model. The conditions for the result to hold are a little relaxed.

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  • Researches on the diversity of growing patterns in biological systems

    Grant number:11214205  1999 - 2001

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research on Priority Areas (B)  Chuo University

    MATSUSHITA Mitsugu, KIMURA Masato, MIMURA Masayasu, KATORI Makoto

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    Grant amount: \13000000 ( Direct Cost: \13000000 )

    Bacteria form characteristic and diverse colonies according to variation of environmental conditions such as nutrient concentration and substrate hardness. This implies that bacteria live actively by interacting collectively or multicellularly with their environment, instead of living unicellularly and passively. In order to elucidate this kind of collective behavior of biological organisms we have studied the growth mechanism and morphological change in colony formation of bacteria both microscopically and macroscopically. We have established morphological diagrams when varying both nutrient concentration and substrate softness. We have also elucidated the growth conditions and characteristics of various colony patterns for species Bacillus subtilis and Proteus mirabilis. Theoretically we have thoroughly investigated how well we can describe our experimental observations from the reaction-diffusion approach. All these results contribute not only to the study of colony formation of bacteria but also to the enlargement of experimental and theoretical studies on pattern formation of the population of biological organisms in general. The species Proteus mirabilis forms macroscopically almost perfect concentric-ring like colonies with approximately equal spacing by repeating collective migration and rest alternately. We have elucidated phenomenological mechanism for the repeated interface growth of the concentric-ring like colonies.

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  • Mathematical Physics

    2001 -  

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    Grant type:Competitive

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  • 森林の林冠ギャップ空間分布の統計物理的研究

    Grant number:10874057  1998 - 2000

    日本学術振興会  科学研究費助成事業  萌芽的研究  中央大学

    香取 眞理

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    Grant amount: \2100000 ( Direct Cost: \2100000 )

    1.森林総研の田中浩氏から提供していただいた小川森林保護区の航空写真による実測データに対して,我々が開発したイジングモデルによる解析方法を適用した.その際,パラメータを決定する新しい方法として,熱平衡状態で成立する相関等式を用いる方法を考案した.実際にこの方法を小川森林保護区の実測データの解析に応用すると,これまでのモンテカルロ法による方法ではパラメータを決定できなかった1991年のデータに対しても,パラメータを決定することが出来た.
    2.小川森林保護区の実測データでは,林冠ギャップの有無という2値のデータだけではなく,各サイトでの樹林の最高値が与えられている.このため,林冠ギャップと林冠とを分けるしきい値を変えた場合に,林冠ギャップ・クラスターの統計性がどのように変わるかを調べることも可能である.従来のモンテカルロ法によるパラメータ探索では,しきい値を変えるごとにモンテカルロ・シミュレーションを行わなければならなかったため,系統的なデータ解析は困難であった.しかし今回,熱平衡相関等式を用いる方法が出来たので,しきい値を変えて系統的に調べることが出来た.結果として,しきい値の選び方に依らず,決定されたパラメータはどれもイジングモデルの臨界値に近いものであった.
    3.森林動態に対する別の数理モデルとして,ある種の森林火災のパーコレーションモデルを提案し,計算機シミュレーションを行なった.そして,森林モデルで臨界性が実現される極限として知られているDrossel-Schwabl極限について考察した.
    4.有効グラフ上のパーコレーションの問題は,森林動態や伝染病伝播の問題の基礎モデルを作る上で重要である.この問題に対して,Arrowsmith-Essam公式と呼ばれるグラフ展開を拡張することが出来た.それにより,いわゆるフレンドリー・ウォークと呼ばれる相互作用する粒子系の統計力学を構成し,この系は高分子や,多成分流体系の「濡れ転移」のモデルと関連深いことを示した.

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  • 1次元粒子系の詳細釣合を満たさない定常分布の解明

    Grant number:06854015  1994    

    日本学術振興会  科学研究費助成事業  奨励研究(A)  中央大学

    香取 眞理

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    Grant amount: \800000 ( Direct Cost: \800000 )

    次のような研究を行った。
    (1)無限に拡がる格子上の無限個の粒子から成る系の定常分布を、これと“dual"な関係にある有限個の粒子から成る系を扱うことによって調べることができる場合がある。我々はGrayによってgraphical representationと呼ばれる方法で導入された、拡張されたdualityを別の方法によって再定式化した。これを応用して、拡張されたcontact processなどいくつかのモデルの臨界値を調べた。
    (2)クラスター間にランダムな競合があるようなdirected percolationを、計算機シミュレーションによって調べた。このような系は、臨界点においてself-affine性をもち、結晶成長系に対して提唱されていたMatsushita-Meakinのスケーリングが成立することが明かとなった。(磯上、松下両氏との共同研究。)
    (3)拡散を伴ったcontact processは、移住の効果も取り入れた伝染病伝播のモデルである。伝染病蔓延⇔撲滅の非平衡相転移の臨界値が、拡散率(移住の速さ)Dにどのように依存するかが問題となる。我々は、空間次元d【greater than or equal】3の場合に、臨界点の上限・下限を与えた。これにより、D→∞やd→∞で、平均場極限にどのように近づくかが明らかになった。crossover現象につ
    (4)非平衡臨界現象が見られるとき、その臨界値や臨界指数は、モデルの時間発展を与えるtransition matrixの固有値の、システムサイズL→∞での漸近形によって定められると考えられる。我々は、coalescing dualという関係にある2つのプロセスは、(各々のパラメーターの間にある関係が成り立つと)transition matrix自体は全く異なっていても、その固有値は任意のLにおいて総て等しくなることを証明した。このことは、coalescing dualの関係にあるプロセスは同じuniversality classに属することを示唆する。(乾、宇沢両氏との共同研究)。

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  • Study of the Ising Models Using Special Purpose Computer System

    Grant number:63302014  1988 - 1990

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Co-operative Research (A)  The University of Tokyo

    SUZUKI Masuo, KATORI Makoto, KATSURA Shigetoshi

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    Grant amount: \10000000 ( Direct Cost: \10000000 )

    The computational physics becomes most important field in modern physics because the computational power has been increasing exponentially and computational study is most effective analyzing method for many important problems. It has been shown that the special purpose computer systems are useful for the theoretical researches. But such systems have not been available unless the researcher constructs one for his study. The main purpose of this research project is to supply such systems to the researchers who need strong simulation power for their studies for the Ising models.
    The special purpose computer system, named m-TIS2, has been developed in this research project. This machine is dedicated for the Monte Carlo simulation of the Ising models and it is the successor to our previous machine, m-TIS, which was developed in 1988. We have constructed thirteen m-TIS2 and have distributed them to the domestic collaborators. Therefore we have achieved the purpose of this project.
    The m-TIS2 has unique architecture. It uses the XILINX's LCA devices. And its hardware configuration can be modified easily by changing the configuration files written into the m-TIS2 at the beginning of the simulation. The fixed circuit is designed so that the m-TIS2 can hold maximum flexibility. Several configuration files are prepared for several models. For example, the CUBIC can simulate the two-dimensional models including up to the third neighbor interactions and three-dimensional models. The SG can treat <plus-minus>J spin-glass models. It is possible to make dedicated machine for different problems if some appropriate configuration files are prepare
    Our m-TIS and m-TIS2 are the first trial for the special purpose computer systems for theoretical sciences in Japan. And our project has been stimulating many other researches in these fields and several similar domestic projects have followed our project.

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  • Probability Theory

    1990 -  

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    Grant type:Competitive

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  • Theoretical Research of Exotic Phase Transitions

    Grant number:63460035  1988 - 1989

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for General Scientific Research (B)  University of Tokyo

    SUZUKI Masuo, KATORI Makoto, MIYASHITA Seiji

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    Grant amount: \6400000 ( Direct Cost: \6400000 )

    The research project "Theoretical Research of Exotic Phase Transitions" was performed by the support of the Grant-in-Aid from the Ministry of Education, Science and Culture during two fiscal years from april 1988.
    A general theory of phase transitions, so-called the coherent-anomaly method (CAM) and the super-effective-field theory have been developed to study exotic phase transitions such as spin glasses and chiral orders. It has been shown that there exists no phase transition in the two-dimensional <plus-minus>J Ising spin glass model, but that there exists a phase transition in three dimensions for the same model. The critical point T_<**> and the nonlinear susceptibility exponent gamma_* have been estimated as T_<**> <similar or equal> 1.4J/k_B and gamma_* <similar or equal> 3. Concerning the scalar chital order, it has been found, using the super-effective-field theory, to be quite difficult to occur in the two-dimensional antiferromagnetic Heisenberg model with cross bonds on a square lattice. This result agrees with other numerical studies.
    Suzuki has discovered a new scheme of "fractal path integrals". This is based on the following general decomposition theorem (Suzuki): e^<x(A+B)> = e^<t1A>e^<t2B>e^<t3A>e^<t4B>...e^<tNA> + O(x^<m+1>), for any positive integer m, where {t_j} are all real numbers proportional to x and they are fractal in their distribution. This new scheme including negative time or temperature is also conceptually interesting and it will be applicable to field theory and nuclear physics. In particular, this mew scheme of fractal decomposition of exponential operators is useful in quantum Monte Carlo simulations, if we combine this new approximant with the Trotter formula, because the resultant error becomes of the order of x^<m+1>n^<-m> for the Trotter number n.
    In order to make effective use of the CAM theory, Suzuki has devised the multi-effective- field theory and the confluent-transfer matrix method. The combination of the CAM theory with the former has given the estimate gamma <similar or equal> 1.7498 for the two-dimensional Ising m

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  • Phase Transitions and Critical Phenomena

    1985 -  

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    Grant type:Competitive

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  • Statistical Physics

    1985 -  

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    Grant type:Competitive

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Teaching Experience

  • 力学

  • 応用解析

  • 電磁気学

  • 量子力学

  • 統計力学

Committee Memberships

  • 2019.4 - 2021.3

    京都大学基礎物理学研究所   共同利用運営委員会議長団  

  • 2019.7 - 2019.12

    the Physical Society of Japan   Selection committee of the 1st Yonezawa Memorial Prize of the Physical Society of Japan  

  • 2009.9 - 2010.8

    日本物理学会   第65期理事  

  • 2008.9 - 2009.8

    日本物理学会   第64期理事  

  • 2001.9 - 2005.9

    日本物理学会   第57-58期代議員  

  • 2001.9 - 2003.8

    日本物理学会   学会誌新著紹介小委員会委員  

  • 2002.5 - 2003.4

    日本物理学会   統計力学・物性基礎論分科会世話人  

  • 1997.11 - 1999.10

    日本物理学会   学会