Favorite sites for simple random walk in two and more dimensions

TL;DR

This paper studies the behavior of favorite sites in simple random walks on multi-dimensional integer lattices, revealing the number and frequency of such sites over time and solving a longstanding open problem.

Contribution

It provides new results on the number of favorite sites in random walks for dimensions two and higher, answering a question posed in 1987.

Findings

In 2D, three favorite sites occur infinitely often.

In 2D, four favorite sites never occur simultaneously.

Sharp asymptotics for the number of favorite sites in dimensions ≥ 3.

Abstract

On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erd\H{o}s and R\'{e}v\'{e}sz (1987), which was brought up again in Dembo (2005).