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Probability and Dynamical Systems Seminar
Probability and Dynamical Systems Seminar
Recurrence and range of a two-dimensional balanced excited random walk
Recurrence and range of a two-dimensional balanced excited random walk
Organizers
Speaker
Time
Tuesday, September 29, 2026 3:15 PM - 4:15 PM
Venue
A3-3-301
Online
Zoom 482 240 1589
(BIMSA)
Abstract
We consider the following walk on $\mathbb{Z}^2$ : Upon the first departure from each vertex, the walk takes a horizontal step; upon every subsequent departure from that vertex, it takes a step of planar simple random walk. Although every step has zero conditional mean, a first departure can increase the distance from the origin more effectively than a simple random walk step. We prove that the walk is recurrent. To our knowledge, this is the first nontrivial recurrence result for the balanced excited walk family. Moreover, we show that the number $R_n$ of distinct vertices visited before time $n$ satisfies $(R_n\log n)/n \to \pi$ almost surely and in every $L^p$, matching the corresponding limit for the planar simple random walk.
The key idea is to use a potential generated by the sites already visited, evaluated at the walker's current position, to control the growth of $R_n$ . Combined with a Lyapunov-function argument, this yields recurrence. These arguments also yield a recurrence criterion for balanced excited walks in cookie environments with suitable positive and negative excitation budgets.
The key idea is to use a potential generated by the sites already visited, evaluated at the walker's current position, to control the growth of $R_n$ . Combined with a Lyapunov-function argument, this yields recurrence. These arguments also yield a recurrence criterion for balanced excited walks in cookie environments with suitable positive and negative excitation budgets.
Speaker Intro
Shuo Qin has been the first Chern Instructor at BIMSA. He obtained a Ph.D. in mathematics in 2024 from New York University under the supervision of Prof. Pierre Tarrès. His work is in probability theory, especially in random processes with memory or reinforcement.