Favorite sites for simple random walk in two and more dimensions

YMSC Probability Seminar

Organizers：

吴昊，杨帆，姜建平，顾陈琳

Speaker：

Yushu Zheng 郑玉书 (AMSS)

Time：

Thur., 3:30-4:30 pm, Sept. 26, 2024

Venue：

C548, Shuangqing Complex Building A

清华大学双清综合楼A座 C548

Title：

Favorite sites for simple random walk in two and more dimensions

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). This talk is based on a joint work with Chenxu Hao, Xinyi Li and Izumi Okada.