Boundary of the Range of Transient Random Walk

TL;DR

This paper investigates the boundary of the range of simple random walks in dimensions three and higher, revealing variance bounds and establishing a central limit theorem for the boundary in higher dimensions.

Contribution

It introduces a martingale-based approach to compare the range and its boundary, providing new variance bounds and a CLT for the boundary in higher dimensions.

Findings

Variance of boundary is of order n log n in dimension 3

Central limit theorem established for the boundary in dimensions 4 and higher

Boundary and range differ mainly by a martingale component

Abstract

We study the boundary of the range of simple random walk on $\mathbb{Z}^d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order $n\log n$ in dimension three. We also establish a central limit theorem in dimension four and larger.