Favorite sites for simple random walk in two and more dimensions

Release time:2025-11-12

DOI number:10.1007/s00440-025-01441-1

Journal:Probability Theory and Related Fields

Key Words:Random walk, Local time, Favorite site

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 (1984), which was brought up again by Dembo (2005).

First Author:Chenxu Hao, Xinyi Li, Izumi Okada, Zheng Yushu

Indexed by:Journal paper

Discipline:Natural Science

First-Level Discipline:Mathematics

Translation or Not:no

Date of Publication:2025-11-12

Included Journals:SCI