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Hideki Tanemura

Publications and source records attributed to Hideki Tanemura.

At least 19 recordsLinked to original sources
arXiv2025

Phase transitions for a unidirectional elephant random walk with a power law memory II: Some sharper estimates

We continue our study of the unidirectional elephant random walk (uERW) initiated in {\it {Electron. Commun. Probab.}} ({\bf 29} 2024, article no. 78). In this paper we obtain definitive results when the memory exponent $β\in (-1, p/(1-p))$. In particular using a coupling argument we obtain the exact asymptotic rate of growth of $S_n$, the location of the uERW at time $n$, for the case $β\in (-1, 0] $. Also, for the case $β\in (0, p/(1-p))$ we show that $P(S_n \to \infty) \in (0,1)$ and conditional on $\{S_n \to \infty\}$ we obtain the exact asymptotic rate of growth of $S_n$. In addition we obtain the central limit theorem for $S_n$ when $β\in (-1, p/(1-p))$.

math.PR↗
arXiv2024

Infinite-dimensional stochastic differential equations arising from Airy random point fields

The Airy$_{β}$ random point fields ($ β= 1,2,4$) are random point fields emerging as the soft-edge scaling limits of eigenvalues of Gaussian random matrices. We construct the unlabeled diffusion reversible with respect to the Airy$_{β}$ random point field for each $ β= 1,2,4$. We identify the infinite-dimensional stochastic differential equations (ISDEs) describing the labeled stochastic dynamics for the unlabeled diffusion mentioned above. We prove the existence and pathwise uniqueness of strong solutions of these ISDEs. Furthermore, the solution of the ISDE is the limit of the solutions of the stochastic differential equations describing the dynamics of the $ N $-particle system in the soft-edge limit. We thus establish the construction of the stochastic dynamics whose unlabeled dynamics are reversible with respect to the Airy random point fields. When $ β=2 $, the solution equals the stochastic dynamics defined by the space-time correlation functions obtained by Prähofer--Spohn, Johansson, Katori--Tanemura, and Corwin--Hammond, among others. We develop a new method whereby these ISDEs have unique, strong solutions. We expect that our approach is valid for other soft-edge scaling limits of stochastic dynamics arising from the random matrix theory.

math.PR↗
arXiv2024

The elephant random walk in the triangular array setting

Gut and Stadmüller (2021, 2022) initiated the study of the elephant random walk with limited memory. Aguech and El Machkouri (2024) published a paper in which they discuss an extension of results by Gut and Stadtmüller (2022) for an "increasing memory" version of the elephant random walk without stops. Here we present a formal definition of the process which has been hinted at Eq. (2.2) in Gut and Stadtmüller (2022). This definition is based on the triangular array setting. We give a positive answer to the open problem in Gut and Stadtmüller (2022) for the elephant random walk, possibly with stops. We also obtain the CLT for the supercritical case of this model.

math.PR↗
arXiv2024

Stochastic differential equations for infinite particle systems of jump type with long range interactions

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a Lévy process, and the interaction between particles is determined by the long-range interaction potential. The potential is of Ruelle's class or logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.

math.PR↗
arXiv2024

Phase transitions for a unidirectional elephant random walk with a power law memory

For the standard elephant random walk, Laulin (2022) studied the case when the increment of the random walk is not uniformly distributed over the past history instead has a power law distribution. We study such a problem for the unidirectional elephant random walk introduced by Harbola, Kumar and Lindenberg (2014). Depending on the memory parameter $p$ and the power law exponent $β$, we obtain three distinct phases in one such phase the elephant travels only a finite distance almost surely, and the other two phases are distinguished by the speed at which the elephant travels.

math.PR↗
arXiv2023

Infinite particle systems with hard-core and long-range interaction

A system of Brownian hard balls is regarded as a reflecting Brownian motion in the configuration space and can be represented by a solution to a Skorohod-type equation. In this article, we consider the case that there are an infinite number of balls, and the interaction between balls is given by the long-range pair interaction. We discuss the existence and uniqueness of strong solutions to the infinite-dimensional Skorohod equation.

math.PR↗
arXiv2021

Infinite-dimensional stochastic differential equations and tail $ σ$-fields II: the IFC condition

In a previous report, the second and third authors gave general theorems for unique strong solutions of infinite-dimensional stochastic differential equations (ISDEs) describing the dynamics of infinitely many interacting Brownian particles. One of the critical assumptions is the \lq\lq IFC" condition. The IFC condition requires that, for a given weak solution, the scheme consisting of the finite-dimensional stochastic differential equations (SDEs) related to the ISDEs exists. Furthermore, the IFC condition implies that each finite-dimensional SDE has unique strong solutions. Unlike other assumptions, the IFC condition is challenging to verify, and so the previous report only verified solution for solutions given by quasi-regular Dirichlet forms. In the present paper, we provide a sufficient condition for the IFC requirement in more general situations. In particular, we prove the IFC condition without assuming the quasi-regularity or symmetry of the associated Dirichlet forms. As an application of the theoretical formulation, the results derived in this paper are used to prove the uniqueness of Dirichlet forms and the dynamical universality of random matrices.

math.PR↗
arXiv2020

Infinite-dimensional stochastic differential equations and tail $ σ$-fields

We present general theorems solving the long-standing problem of the existence and pathwise uniqueness of strong solutions of infinite-dimensional stochastic differential equations (ISDEs) called interacting Brownian motions. These ISDEs describe the dynamics of infinite-many Brownian particles moving in $ \mathbb{R}^d $ with free potential $ Φ$ and mutual interaction potential $ Ψ$. We apply the theorems to essentially all interaction potentials of Ruelle's class such as the Lennard-Jones 6-12 potential and Riesz potentials, and to logarithmic potentials appearing in random matrix theory. We solve ISDEs of the Ginibre interacting Brownian motion and the sine$_β$ interacting Brownian motion with $ β= 1,2,4$. We also use the theorems in separate papers for the Airy and Bessel interacting Brownian motions. One of the critical points for proving the general theorems is to establish a new formulation of solutions of ISDEs in terms of tail $ σ$-fields of labeled path spaces consisting of trajectories of infinitely many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite dimensions.

math.PR↗
arXiv2020

On a model of evolution of subspecies

Ben-Ari and Schinazi (2016) introduced a stochastic model to study `virus-like evolving population with high mutation rate'. This model is a birth and death model with an individual at birth being either a mutant with a random fitness parameter in $[0,1]$ or having one of the existing fitness parameters with uniform probability; whereas a death event removes the entire population of the least fit site. We change this to incorporate the notion of `survival of the fittest', by requiring that a non-mutant individual, at birth, has a fitness according to a preferential attachment mechanism, i.e., it has a fitness $f$ with a probability proportional to the size of the population of fitness $f$. Also death just removes one individual at the least fit site. This preferential attachment rule leads to a power law behaviour in the asymptotics, unlike the exponential behaviour obtained by Ben-Ari and Schinazi (2016).

math.PR↗
arXiv2019

Uniqueness of Dirichlet forms related to infinite systems of interacting Brownian motions

The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field $ μ$, there exist two natural infinite-volume Dirichlet forms $ (\mathcal{E}^{\mathsf{upr}},\mathcal{D}^{\mathsf{upr}})$ and $(\mathcal{E}^{\mathsf{lwr}},\mathcal{D}^{\mathsf{lwr}})$ on $ L^2(\mathsf{S} ,μ) $ describing interacting Brownian motions each with unlabeled equilibrium state $ μ$. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by $(\mathcal{E}^{\mathsf{lwr}},\mathcal{D}^{\mathsf{lwr}})$ is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of {\ws}s of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle's class interaction potentials, and Theorem 3.2 to the sine$ _2$ interacting Brownian motion and interacting Brownian motions with Ruelle's class interaction potentials of $ C_0^3 $-class.

math.PR↗
arXiv2016

Isotropic-nematic behaviour of hard rigid rods: a percolation theoretic approach

Needles at different orientations are placed in an i.i.d. manner at points of a Poisson point process on $\mathbb R^2$ of density $λ$. Needles at the same direction have the same length, while needles at different directions maybe of different lengths. We study the geometry of a finite cluster when needles have only two possible orientations and when needles have only three possible orientations. In both these cases the asymptotic shape of the finite cluster as $λ\to \infty$ is shown to consists of needles only in two directions. In the two orientations case the shape does not depend on the orientation but just on the i.i.d. structure of the orientations, while in the three orientations case the shape depend on all the parameters, i.e. the i.i.d. structure of the orientations, the lengths and the orientations of the needles.

math-ph↗
arXiv2015

Strong Markov property of determinantal processes with extended kernels

Noncolliding Brownian motion (Dyson's Brownian motion model with parameter $β=2$) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes.

math.PR↗
arXiv2014

Cores of Dirichlet forms related to random matrix theory

We prove the sets of polynomials on configuration spaces are cores of Dirichlet forms describing interacting Brownian motion in infinite dimensions. Typical examples of these stochastic dynamics are Dyson's Brownian motion and Airy interacting Brownian motion. Both particle systems have logarithmic interaction potentials, and naturally arise from random matrix theory. The results of the present paper will be used in a forth coming paper to prove the identity of the infinite-dimensional stochastic dynamics related to the random matrix theories constructed by apparently different methods: the method of space-time correlation functions and that of stochastic analysis.

math.PR↗
arXiv2013

Complex Brownian Motion Representation of the Dyson Model

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.

math.PR↗
arXiv2011

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

When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $ξ=\sum_{j \in Λ} δ_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $Φ$ having zeros in $\supp ξ$. Using such entire functions $Φ$, we define new topologies called the $Φ$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, as the limits of sequences of determinantal diffusion processes with finite numbers of particles in the sense of finite dimensional distributions in the $Φ$-moderate topologies, so that the probability distributions are continuous with respect to initial configurations $ξ$ with $ξ(\R)=\infty$. We show that our three infinite particle systems are versions of the determinantal processes with the extended sine, Bessel, and Airy kernels, respectively, which are reversible with respect to the determinantal point processes obtained in the bulk scaling limit and the soft-edge scaling limit of the eigenvalue distributions of the Gaussian unitary ensemble, and the hard-edge scaling limit of that of the chiral Gaussian unitary ensemble studied in the random matrix theory. Then Markovianity is proved for the three infinite-dimensional determinantal processes.

math.PR↗
arXiv2011

Noncolliding processes, matrix-valued processes and determinantal processes

A noncolliding diffusion process is a conditional process of $N$ independent one-dimensional diffusion processes such that the particles never collide with each other. This process realizes an interacting particle system with long-ranged strong repulsive forces acting between any pair of particles. When the individual diffusion process is a one-dimensional Brownian motion, the noncolliding process is equivalent in distribution with the eigenvalue process of an $N \times N$ Hermitian-matrix-valued process, which we call Dyson's model. For any deterministic initial configuration of $N$ particles, distribution of particle positions of the noncolliding Brownian motion on the real line at any fixed time $t >0$ is a determinantal point process. We can prove that the process is determinantal in the sense that the multi-time correlation function for any chosen series of times, which determines joint distributions at these times, is also represented by a determinant. We study the asymptotic behavior of the system, when the number of Brownian motions $N$ in the system tends to infinity. This problem is concerned with the random matrix theory on the asymptotics of eigenvalue distributions, when the matrix size becomes infinity. In the present paper, we introduce a variety of noncolliding diffusion processes by generalizing the noncolliding Brownian motion, some of which are temporally inhomogeneous. We report the results of our research project to construct and study finite and infinite particle systems with long-ranged strong interactions realized by noncolliding processes.

math.PR↗