Year: 2026
Type: preprint
Abstract: We prove that the planar balanced excited random walk $M(2,1,2)$ is recurrent. This walk takes a horizontal simple random walk step on its first departure from each vertex and a planar simple random walk step on every later departure. Moreover, the number of distinct vertices visited before time $n$, multiplied by $(\log n)/n$, converges to $π$ almost surely and in every $L^p$, $1\le p<\infty$, the same limit as for the planar simple random walk. More generally, we prove recurrence of balanced excited random walks in spatially inhomogeneous cookie environments whenever the total positive and negative cookie strengths at each vertex are bounded by constants $A, B$ with $A+B<1+1/(2π+1)$.
Source: arXiv (Cornell University)
Author: Qin, Shuo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214384274