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arXiv · 2409.00995

Favorite sites for simple random walk in two and more dimensions

Chenxu Hao·Xinyi Li·Izumi Okada·Yushu Zheng
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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 Erdős and Révész (1987), which was brought up again in Dembo (2005).

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Chenxu Hao, Xinyi Li, Izumi Okada, Yushu Zheng. 2025-11-12. Favorite sites for simple random walk in two and more dimensions. https://doi.org/10.1007/s00440-025-01441-1

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