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

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 ${\mathbb{Z}}^d$ $(d\ge 3)$, 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 $d\ge 2$, where ${\beta }_d$ is a certain constant having a simple probabilistic expression.

Accepted for the publication in Stochastic processes and their applications

References in corpus (1)

• Late points for random walks in two dimensions

Cited by in corpus (2)

• Exceptional points of two-dimensional random walks at multiples of the cover time
• Maximum and minimum of local times for two-dimensional random walk