The inner boundary of random walk range

TL;DR

This paper investigates the properties of the inner boundary of a random walk range, establishing almost sure limits, large deviation results, and asymptotic expectations for two-dimensional simple random walks.

Contribution

It proves the almost sure existence of the limit of the normalized inner boundary points and provides asymptotic and large deviation results for the inner boundary of random walks.

Findings

Limit of $L_n/n$ exists with probability one.

Expectation of inner boundary points is of order $n/( ext{log } n)^2$.

Large deviation results for transient walks.

Abstract

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 $\frac{n}{(\log n)^2}$.