The asymptotic behavior of rarely visited edges of the simple random walk

Ze-Chun Hu, Xue Peng, Renming Song, Yuan Tan

Published 2023-10-25Version 1

In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(0,+\infty)$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely.