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

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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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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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

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

DOI： 10.1063/1.5145357

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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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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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Language：English   Publishing type：Research paper (scientific journal)  

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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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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Language：English   Publishing type：Research paper (scientific journal)  

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.

PubMed

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

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：NATL ACAD SCI UKRAINE, INST MATH  

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.

DOI： 10.3842/SIGMA.2017.079

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

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.

DOI： 10.1063/1.4964253

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Language：Japanese   Publisher：The Physical Society of Japan  

DOI： 10.11316/jpsgaiyo.71.2.0_2790

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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.

DOI： 10.7566/JPSJ.84.124001

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Language：English   Publisher：Kyoto University  

CiNii Books

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Other Link： http://hdl.handle.net/2433/224305

Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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.

DOI： 10.7566/JPSJ.84.114002

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

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.

DOI： 10.1007/s00340-015-6130-0

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

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.

DOI： 10.1007/s00440-014-0581-9

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

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.

DOI： 10.1007/s10955-014-1179-4

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

DOI： 10.11316/jpsgaiyo.70.2.0_2755

CiNii Books

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

DOI： 10.11316/jpsgaiyo.70.2.0_2839

CiNii Books

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

DOI： 10.11316/jpsgaiyo.70.2.0_2866

CiNii Books

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

DOI： 10.11316/jpsgaiyo.70.2.0_2867

CiNii Books

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

DOI： 10.11316/jpsgaiyo.70.1.0_3214

CiNii Books

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

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.

DOI： 10.1016/j.spa.2014.06.002

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

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.

DOI： 10.1063/1.4894235

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Institute of Physics Publishing  

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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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.

DOI： 10.1214/ECP.v18-2554

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

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.

DOI： 10.1007/s10955-012-0602-y

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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.

DOI： 10.1088/1751-8113/45/39/395201

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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]

DOI： 10.1063/1.4758795

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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.

DOI： 10.1007/s10955-012-0527-5

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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).

DOI： 10.1007/s10955-012-0472-3

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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).

DOI： 10.1007/s10955-011-0372-y

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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.

DOI： 10.1103/PhysRevE.84.061144

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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.

DOI： 10.1103/PhysRevE.84.041141

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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.

DOI： 10.1143/JPSJ.80.074003

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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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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.

DOI： 10.1103/PhysRevE.83.041127

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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.

DOI： 10.1007/s10955-011-0117-y

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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.

DOI： 10.1103/PhysRevA.81.012314

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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.

DOI： 10.1007/s00220-009-0912-3

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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)).

DOI： 10.1103/PhysRevA.81.012314

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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.

DOI： 10.1007/s10955-009-9829-7

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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.

DOI： 10.1103/PhysRevE.78.051102

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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.

DOI： 10.1007/s10955-008-9524-0

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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.

DOI： 10.1103/PhysRevA.77.062331

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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.

DOI： 10.1007/s10955-007-9421-y

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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.

DOI： 10.1103/PhysRevA.76.012332

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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.

DOI： 10.1103/PhysRevA.76.012332

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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.

DOI： 10.1007/s00440-006-0015-4

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DOI： 10.1103/PhysRevA.72.019904

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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.

DOI： 10.1103/PhysRevA.72.012316

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

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.

DOI： 10.1063/1.1765215

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC/PLENUM PUBL  

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).

DOI： 10.1023/B:JOTP.0000020478.24536.26

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER PHYSICAL SOC  

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.

DOI： 10.1103/PhysRevE.68.021112

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER PHYSICAL SOC  

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.

DOI： 10.1103/PhysRevE.67.051110

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

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：IOP PUBLISHING LTD  

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.

DOI： 10.1088/0305-4470/36/3/302

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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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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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Language：Japanese   Publisher：The Japan Society for Industrial and Applied Mathematics  

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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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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER PHYSICAL SOC  

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.

DOI： 10.1103/PhysRevE.66.011105

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：INST MATHEMATICAL STATISTICS  

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.

DOI： 10.1214/aop/1023481012

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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.

DOI： 10.1143/JPSJ.70.359

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

DOI： 10.11316/butsuri1946.55.276

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Other Link： https://jlc.jst.go.jp/DN/JALC/00065983461?from=CiNii

Language：English   Publishing type：Research paper (scientific journal)   Publisher：SOCIEDADE BRASILEIRA FISICA  

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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DOI： 10.1023/A:1018617328216

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMERICAN PHYSICAL SOC  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOCIETY JAPAN  

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.

DOI： 10.1143/JPSJ.68.2553

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier Science Publishers B.V.  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Elsevier Science Publishers B.V.  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：IOP PUBLISHING LTD  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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)).

DOI： 10.1143/JPSJ.67.1655

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

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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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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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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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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PLENUM PUBL CORP  

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 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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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.

DOI： 10.1143/JPSJ.65.2536

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：IOP PUBLISHING LTD  

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.

DOI： 10.1088/0305-4470/29/15/010

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

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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DOI： 10.1103/PhysRevB.54.R9620

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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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：IOP PUBLISHING LTD  

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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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.

DOI： 10.1088/0305-4470/28/7/007

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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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOC JAPAN  

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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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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PROGRESS THEORETICAL PHYSICS PUBLICATION OFFICE  

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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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.

DOI： 10.1088/0305-4470/26/23/011

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

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.

DOI： 10.1016/0378-4371(92)90218-F

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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.

DOI： 10.1143/JPSJ.61.806

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

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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PLENUM PUBL CORP  

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.

DOI： 10.1007/BF01329859

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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.

DOI： 10.1016/0378-4371(91)90344-C

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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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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOCIETY JAPAN KIKAI-SHINKO BUILDING  

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.

DOI： 10.1143/JPSJ.60.418

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：PHYSICAL SOCIETY JAPAN KIKAI-SHINKO BUILDING  

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).

DOI： 10.1143/JPSJ.60.430

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DOI： 10.1142/S0217984991000198

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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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DOI： 10.1143/JPSJ.59.2681

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DOI： 10.1143/JPSJ.59.1581

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DOI： 10.1143/JPSJ.59.877

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DOI： 10.1143/JPSJ.57.4114

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DOI： 10.1143/JPSJ.57.3753

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DOI： 10.1143/JPSJ.57.807

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DOI： 10.1143/JPSJ.56.3865

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DOI： 10.1143/JPSJ.56.3092

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DOI： 10.1143/JPSJ.56.3113

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DOI： 10.1143/JPSJ.55.1

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KYOTO UNIV  

DOI： 10.1143/PTP.74.1175

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

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Language：English   Publishing type：Article, review, commentary, editorial, etc. (international conference proceedings)   Publisher：SPRINGER-VERLAG TOKYO  

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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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

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

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

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Language：Japanese   Publishing type：Book review, literature introduction, etc.  

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

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Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)   Publisher：サイエンス社  

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Language：English   Publishing type：Rapid communication, short report, research note, etc. (bulletin of university, research institution)  

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Language：English   Publishing type：Rapid communication, short report, research note, etc. (bulletin of university, research institution)  

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Language：English   Publishing type：Rapid communication, short report, research note, etc. (bulletin of university, research institution)  

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Language：English   Publishing type：Article, review, commentary, editorial, etc. (international conference proceedings)   Publisher：PLENUM PRESS DIV PLENUM PUBLISHING CORP  

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

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Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

DOI： 10.11316/butsuri1946.49.100

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Language：English   Publishing type：Article, review, commentary, editorial, etc. (international conference proceedings)  

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Language：Japanese   Publisher：The Physical Society of Japan (JPS)  

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