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Hideki Tanemura

Publications and source records attributed to Hideki Tanemura.

33 records · Page 2Linked to original sources
arXiv2011

Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index $ν> -1$ conditioned never to collide with each other, in which if $-1 < ν< 0$ 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 $J_ν$ 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.

math.PR↗
arXiv2009

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson's model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is $β/2$ times the inverse of particle distance. When $β=2$, 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's model with $β=2$ and $N$ particles, $\X(t)=(X_1(t), ..., X_N(t)), t \in [0,\infty), 2 \leq N < \infty$, 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's model with $β=2$ starting from the first $N$ zeros of $\Ai(z)$, $0 > a_1 > ... > a_N$, $N \geq 2$. 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 : $Y_j(t) = X_j(t) + t^2/4 + \{d_1 + \sum_{\ell=1}^N (1/a_{\ell})\}t, 1 \leq j \leq N$, where $d_1=\Ai'(0)/\Ai(0)$. We show that, as the $N \to \infty$ limit of $\Y(t)=(Y_1(t), ..., Y_N(t)), t \in [0, \infty)$, 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 $μ_{\Ai}$. The stationary state $μ_{\Ai}$ 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.

math.PR↗
arXiv2009

Non-Equilibrium Dynamics of Dyson's Model with an Infinite Number of Particles

Dyson'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 $β/2$. We give sufficient conditions for initial configurations so that Dyson's model with $β=2$ 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.

math.PR↗
arXiv2007

Noncolliding Brownian Motion and Determinantal Processes

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.

math.PR↗
arXiv2006

Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals

Yor's generalized meander is a temporally inhomogeneous modification of the $2(ν+1)$-dimensional Bessel process with $ν> -1$, in which the inhomogeneity is indexed by $κ\in [0, 2(ν+1))$. We introduce the non-colliding particle systems of the generalized meanders and prove that they are the 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 $J_ν$ used in the fractional calculus, where orders of differintegration are determined by $ν-κ$. As special cases of the two parameters $(ν, κ)$, 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.

math.PR↗
arXiv2005

Nonintersecting Paths, Noncolliding Diffusion Processes and Representation Theory

The system of one-dimensional symmetric simple random walks, in which none of walkers have met others in a given time period, is called the vicious walker model. It was introduced by Michael Fisher and applications of the model to various wetting and melting phenomena were described in his Boltzmann medal lecture. In the present report, we explain interesting connections among representation theory, probability theory, and random matrix theory using this simple diffusion particle system. Each vicious walk of $N$ walkers is represented by an $N$-tuple of nonintersecting lattice paths on the spatio-temporal plane. There is established a simple bijection between nonintersecting lattice paths and semistandard Young tableaux. Based on this bijection and some knowledge of symmetric polynomials called the Schur functions, we can give a determinantal expression to the partition function of vicious walks, which is regarded as a special case of the Karlin-McGregor formula in the probability theory (or the Lindström-Gessel-Viennot formula in the enumerative combinatorics). Due to a basic property of Schur function, we can take the diffusion scaling limit of the vicious walks and define a noncolliding system of Brownian particles. This diffusion process solves the stochastic differential equations with the drift terms acting as the repulsive two-body forces proportional to the inverse of distances between particles, and thus it is identified with Dyson's Brownian motion model. In other words, the obtained noncolliding system of Brownian particles is equivalent in distribution with the eigenvalue process of a Hermitian matrix-valued process.

math.PR↗
arXiv2004

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

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.

math-ph↗
arXiv2004

Functional central limit theorems for vicious walkers

We consider the diffusion scaling limit of the vicious walker model that is a system of nonintersecting random walks. We prove a functional central limit theorem for the model and derive two types of nonintersecting Brownian motions, in which the nonintersecting condition is imposed in a finite time interval $(0,T]$ for the first type and in an infinite time interval $(0,\infty)$ for the second type, respectively. The limit process of the first type is a temporally inhomogeneous diffusion, and that of the second type is a temporally homogeneous diffusion that is identified with a Dyson's model of Brownian motions studied in the random matrix theory. We show that these two types of processes are related to each other by a multi-dimensional generalization of Imhof's relation, whose original form relates the Brownian meander and the three-dimensional Bessel process. We also study the vicious walkers with wall restriction and prove a functional central limit theorem in the diffusion scaling limit.

math.PR↗
arXiv2003

The structure of finite clusters in high intensity Poisson Boolean stick process

Sticks at one of different orientation are placed in an i.i.d. fashion at points of a Poisson point process of intensity $λ$. Sticks of the same direction have the same length, while sticks in different directions may have different lengths. We study the geometry of finite cluster as $λ\to \infty$. The asymptotic shape of the custer being determined by the probabilities of the sticks in various direction and their lengths and orientations. We also obtain the limiting geometric structure of this component.

math.PR↗
arXiv2003

Noncolliding Brownian motions and Harish-Chandra formula

We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a finite time interval $(0,T]$. This is a temporally inhomogeneous diffusion process whose transition probability density depends on a value of $T$, and in the limit $T \to \infty$ it converges to a temporally homogeneous diffusion process called Dyson's model of Brownian motions. It is known that the distribution of particle positions in Dyson's model coincides with that of eigenvalues of a Hermitian matrix-valued process, whose entries are independent Brownian motions. In the present paper we construct such a Hermitian matrix-valued process, whose entries are sums of Brownian motions and Brownian bridges given independently of each other, that its eigenvalues are identically distributed with the particle positions of our temporally inhomogeneous system of noncolliding Brownian motions. As a corollary of this identification we derive the Harish-Chandra formula for an integral over the unitary group.

math.PR↗
arXiv2003

Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices

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 a motion of radial coordinate of the three-dimensional Brownian motion represented in the 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-deGennes Hamiltonians of the mean-field theory of superconductivity, which have the particle-hole symmetry. We report that the time evolution of the present stochastic process is fully characterized by the change of symmetry classes from the type $C$ to the type $C$I in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the non-colliding 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.

cond-mat.stat-mech↗
arXiv2003

Infinite systems of non-colliding Brownian particles

Non-colliding Brownian particles in one dimension is studied. $N$ Brownian particles start from the origin at time 0 and then they do not collide with each other until finite time $T$. We derive the determinantal expressions for the multitime correlation functions using the self-dual quaternion matrices. We consider the scaling limit of the infinite particles $N \to \infty$ and the infinite time interval $T \to \infty$. Depending on the scaling, two limit theorems are proved for the multitime correlation functions, which may define temporally inhomogeneous infinite particle systems.

math.PR↗
arXiv2002

Scaling limit of vicious walks and two-matrix model

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 $N \times N$ 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.

cond-mat.stat-mech↗
arXiv2002

Dynamical Correlations among Vicious Random Walkers

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 $t \ll T$, 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). The Brownian motion is generally described by the dynamical correlations among particles at many times $t_1,t_2,..., t_M$ between t=0 and t=T. 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 $N \to \infty$ are evaluated and a discontinuous transition of the universality class from GUE to GOE is observed.

cond-mat.stat-mech↗