On Escaping, Entering, and Visiting Discs of Projections of Planar Symmetric Random Walks on the Lattice Torus

TL;DR

This paper extends the analysis of escape, entrance times, and hitting distributions for symmetric random walks on the lattice torus, providing new insights into their behavior using purely random walk methods.

Contribution

It generalizes previous frameworks to a broader class of planar random walks on the lattice torus, focusing on escape and entrance properties.

Findings

Derived bounds for escape and entrance times

Extended Green's function estimates to lattice tori

Analyzed hitting distributions on discs and annuli

Abstract

We examine escape and entrance times, Green's functions, local times, and hitting distributions of discs and annuli of a symmetric random walk on $\Z^2$ projected onto the periodic lattice $\Z^2_K$. This extends a framework for the simple planar random walk in Dembo, et al. (2006) to the large class of planar random walks in Bass, Rosen (2007). The approach uses comparisons between $\Z^2$ and $\Z^2_K$ hitting times and distributions on annuli, and uses only random walk methods.