arXiv subjects

Yoshihiro Abe

Publications and source records attributed to Yoshihiro Abe.

8 recordsLinked to original sources

arXiv2025

Extremal process of the local time of simple random walk on a regular tree

We study a continuous-time simple random walk on a regular rooted tree of depth $n$ in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has spent a prescribed amount of time there. In both cases we show that the extremal process associated with centered square-root local time on the leaves tends, as $n\to\infty$, to a decorated Poisson point process with a random intensity measure. While the intensity measure is specific to the local-time problem at hand, the decorations are exactly those for the tree-indexed Markov chain (a.k.a. Branching Random Walk or Gaussian Free Field) with normal step distribution. The proof demonstrates the latter by way of a Lindeberg-type swap of the decorations of the two processes which itself relies on a well-known isomorphism theorem.

math.PR↗

arXiv2022

Exceptional points of two-dimensional random walks at multiples of the cover time

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq \mathbb Z^2$ of bounded open domains $D\subseteq \mathbb R^2$. Upon exit from $D_N$, the walk lands on a "boundary vertex" and then reenters $D_N$ through a random boundary edge in the next step. In the parametrization by the local time at the "boundary vertex" we prove that, at times corresponding to a $θ$-multiple of the cover time of $D_N$, the sets of suitably defined $λ$-thick (i.e., heavily visited) and $λ$-thin (i.e., lightly visited) points are, as $N\to\infty$, distributed according to the Liouville Quantum Gravity $Z^D_λ$ with parameter $λ$-times the critical value. For $θ<1$, also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to $Z^D_{\sqrtθ}$. The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.

math.PR↗

arXiv2020

Second order term of cover time for planar simple random walk

We consider the cover time for a simple random walk on the two-dimensional discrete torus of side length $n$. Dembo, Peres, Rosen, and Zeitouni [Ann. Math. 160:433-464, 2004] identified the leading term in the asymptotics for the cover time as $n$ goes to infinity. In this paper, we study the exact second order term. This is a discrete analogue of the work on the cover time for planar Brownian motion by Belius and Kistler [Probab. Theory Relat Fields. 167:461-552, 2017].

math.PR↗

arXiv2019

Exceptional points of discrete-time random walks in planar domains

Given a sequence of lattice approximations $D_N\subset\mathbb Z^2$ of a bounded continuum domain $D\subset\mathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $\varrho$, we consider discrete-time simple random walks in $D_N\cup\{\varrho\}$ run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in $D$. The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at $\varrho$. A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.

math.PR↗

arXiv2017

Extremes of local times for simple random walks on symmetric trees

We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim_{n \to \infty} r_n = \infty$ and $\lim_{n \to \infty} r_n/n \in [0, 1)$. We show that the point process weakly converges to a Cox process with intensity measure $αZ_{\infty} (dx) \otimes e^{-2\sqrt{\log b}~y}dy$, where $α> 0$ is a constant and $Z_{\infty}$ is a random measure on $[0, 1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.

math.PR↗

arXiv2014

Maximum and minimum of local times for two-dimensional random walk

We obtain the leading orders of the maximum and the minimum of local times for the simple random walk on the two-dimensional torus at time proportional to the cover time. We also estimate the number of points with large (or small) values of the local times. These are analogues of estimates on the two-dimensional Gaussian free fields by Bolthausen, Deuschel, and Giacomin [Ann. Probab., 29 (2001)] and Daviaud [Ann. Probab., 34 (2006)], but we have different exponents from the case of the Gaussian free field.

math.PR↗

arXiv2013

Effective resistances for supercritical percolation clusters in boxes

Let $\mathcal{C}^n$ be the largest open cluster for supercritical Bernoulli bond percolation in $[-n, n]^d \cap \mathbb{Z}^d$ with $d \ge 2$. We obtain a sharp estimate for the effective resistance on $\mathcal{C}^n$. As an application we show that the cover time for the simple random walk on $\mathcal{C}^n$ is comparable to $n^d (\log n)^2$. Noting that the cover time for the simple random walk on $[-n, n]^d \cap \mathbb{Z}^d$ is of order $n^d \log n$ for $d \ge 3$ (and of order $n^2 (\log n)^2$ for $d = 2$), this gives a quantitative difference between the two random walks for $d \ge 3$.

math.PR↗

arXiv2013

Cover times for sequences of reversible Markov chains on random graphs

We provide conditions that classify cover times for sequences of random walks on random graphs into two types: One type (Type 1) is the class of cover times that are of the order of the maximal hitting times scaled by the logarithm of the size of vertex sets. The other type (Type 2) is the class of cover times that are of the order of the maximal hitting times. The conditions are described by some parameters determined by the underlying graphs: the volumes, the diameters with respect to the resistance metric, the coverings or packings by balls in the resistance metric. We apply the conditions to and classify a number of examples, such as supercritical Galton-Watson trees, the incipient infinite cluster of a critical Galton-Watson tree and the Sierpinski gasket graph.

math.PR↗