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Izumi Okada

Publications and source records attributed to Izumi Okada.

15 recordsLinked to original sources
arXiv2026

Graph distance and effective resistance of the random walk trace in four and five dimensions

In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distribution in dimensions six and higher. Taken together, these results reveal a phase transition between dimensions five and six. Our proof develops a novel coupling with long range percolation, and we expect this technique to find applications in a broad class of related models.

math.PR↗
arXiv2026

Phase transition on the fluctuation of the structure of random walk ranges

We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to qualitatively different regimes, with a distinct phase transition in dimension 6. In particular, we remark that convergence in dimension 6 occurs with a non-standard rescaling.

math.PR↗
arXiv2026

Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints

We discuss the relationship between the capacity and the geometry for the range of the random walk for $d=3$. In particular, we consider how efficiently the random walk moves or what shape it forms in order to maximize its capacity. In one of our main results, we show a functional law for the capacity of the random walk. In addition, we find that there is a phase transition for the asymptotics of the capacity of the random walk when we condition the diameter of the random walk.

math.PR↗
arXiv2025

Favorite sites for simple random walk in two and more dimensions

On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987), which was brought up again in Dembo (2005).

math.PR↗
arXiv2024

Moderate Deviations for the Capacity of the Random Walk range in dimension four

In this paper, we find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen \cite{BCR} concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, \begin{equation*} \label{eq:maxineq} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{1/2}_{L^2} \|\nabla f\|^{1/2}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation*}

math.PR↗
arXiv2024

Capacity of the range of random walk: The law of the iterated logarithm

We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk.

math.PR↗
arXiv2023

Branching random walks and Minkowski sum of random walks

We show that the range of a critical branching random walk conditioned to survive forever and the Minkowski sum of two independent simple random walk ranges are intersection-equivalent in any dimension $d\ge 5$, in the sense that they hit any finite set with comparable probability, as their common starting point is sufficiently far away from the set to be hit. Furthermore, we extend a discrete version of Kesten, Spitzer and Whitman's result on the law of large numbers for the volume of a Wiener sausage. Here, the sausage is made of the Minkowski sum of $N$ independent simple random walk ranges in $\mathbb{Z}^d$, with $d>2N$, and of a finite set $A\subset \mathbb{Z}^d$. When properly normalised the volume of the sausage converges to a quantity equivalent to the capacity of $A$ with respect to the kernel $K(x,y)=(1+\|x-y\|)^{2N-d}$. As a consequence, we establish a new relation between capacity and {\it branching capacity}.

math.PR↗
arXiv2023

Deviations of the intersection of Brownian Motions in dimension four with general kernel

In this paper, we find a natural four dimensional analog of the moderate deviation results of Chen (2004) for the mutual intersection of two independent Brownian motions $B$ and $B'$. In this work, we focus on understanding the following quantity, for a specific family of kernels $H$, \begin{equation*} \int_0^1 \int_0^1 H (B_s - B'_t) \text{d}t \text{d}s . \end{equation*} Given $H(z) \propto \frac{1}{|z|^γ}$ with $0 < γ\le 2$, we find that the deviation statistics of the above quantity can be related to the following family of inequalities from analysis, \begin{equation} \label{eq:maxineq} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{(1-γ/4)}_{L^2} \|\nabla f\|^{γ/4}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) H(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation} Furthermore, in the case that $H$ is the Green's function, the above will correspond to the generalized Gagliardo-Nirenberg inequality; this is used to analyze the Hartree equation in the field of partial differential equations. Thus, in this paper, we find a new and deep link between the statistics of the Brownian motion and a family of relevant inequalities in analysis.

math.PR↗
arXiv2020

Isolated singularities in the heat equation behaving like fractional Brownian motions

We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent $α\in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $α$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2] $. It turns out that $H=1/N$ is critical.

math.AP↗
arXiv2019

Geometric structures of late points of a two-dimensional simple random walk

We consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite points and high points in the Gaussian free field, whose exact values are known. We estimate the exponents for the numbers of a multipoint set of late points in average. While there have been observed certain similarities between among three classes of points, our results exhibit a difference.

math.PR↗
arXiv2016

Topics and problems on favorite sites of random walks

In this article, we study special points of a simple random walk and a Gaussian free field, such as (nearly) favorite points, late points and high points. In section $2$, we extend results of [19] and suggest open problems for $d=2$. In section $3$, we give a survey on the geometric structures of (nearly) favorite points and late points of a simple random walk and high points of a Gaussian free field in two dimension.

math.PR↗
arXiv2016

Frequently visited sites of the inner boundary of simple random walk range

This paper considers the question: how many times does a simple random walk revisit the most frequently visited site among the inner boundary points? It is known that in ${\mathbb{Z}}^2$, the number of visits to the most frequently visited site among all of the points of the random walk range up to time $n$ is asymptotic to $π^{-1}(\log n)^2$, while in ${\mathbb{Z}}^d$ $(d\ge3)$, it is of order $\log n$. We prove that the corresponding number for the inner boundary is asymptotic to $β_d\log n$ for any $d\ge2$, where $β_d$ is a certain constant having a simple probabilistic expression.

math.PR↗
arXiv2014

The inner boundary of random walk range

In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If $L_n$ be the number of the inner boundary points of random walk range in the $n$ steps, we prove $\lim_{n\to \infty}\frac{L_n}{n}$ exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on two dimensionnal square lattice is of the same order as $\frac{n}{(\log n)^2}$.

math.PR↗
arXiv2014

Last zero time or Maximum time of the winding number of Brownian motions

In this paper we consider the winding number, $θ(s)$, of planar Brownian motion and study asymptotic behavior of the process of the maximum time, the time when $θ(s)$ attains the maximum in the interval $0\le s \le t$. We find the limit law of its logarithm with a suitable normalization factor and the upper growth rate of the maximum time process itself. We also show that the process of the last zero time of $θ(s)$ in $[0,t]$ has the same law as the maximum time process.

math.PR↗