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

TL;DR

This paper investigates the frequency of revisits to the most visited site on the inner boundary of a simple random walk range, revealing asymptotic behaviors across different dimensions.

Contribution

It establishes the asymptotic number of revisits to the most visited inner boundary site for all dimensions, extending known results to boundary points.

Findings

In ${f Z}^2$, revisits grow as $( ext{constant}) imes ( ext{log } n)^2$.

In ${f Z}^d$, $d extgreater 2$, revisits grow linearly with $ ext{log } n$.

The asymptotic constant $eta_d$ is explicitly characterized.

Abstract

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 $\pi^{-1}(\log n)^2$, 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 $\beta_d\log n$ for any $d\ge2$, where $\beta_d$ is a certain constant having a simple probabilistic expression.