Fluctuation results for size of the vacant set for random walks on discrete torus

TL;DR

This paper investigates the fluctuation behavior of the size of certain random sets, like vacant sets, generated by independent random walks on high-dimensional discrete tori, providing refined tail estimates and broad applicability.

Contribution

It offers new fluctuation results for the size of vacant sets and intersections in random walks on discrete tori, with a refined analysis of hitting time tail estimates.

Findings

Fluctuation behavior characterized for vacant sets and intersections.

Refined tail estimates for hitting times on discrete tori.

Results applicable to other vertex-transitive graphs.

Abstract

We consider one or more independent random walks on the $d\ge 3$ dimensional discrete torus. The walks start from vertices chosen independently and uniformly at random. We analyze the fluctuation behavior of the size of some random sets arising from the trajectories of the random walks at a time proportional to the size of the torus. Examples include vacant sets and the intersection of ranges. The proof relies on a refined analysis of tail estimates for hitting time and can be applied for other vertex-transitive graphs.