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

TL;DR

This paper analyzes the rarely visited edges of a one-dimensional simple random walk by studying $\alpha(n)$, the number of edges visited exactly once up to time $n$. It first proves $\mathbb{E}[\alpha(n)]$ is nondecreasing with $\mathbb{E}[\alpha(n)] \to 2$ as $n\to\infty$, revealing a contrast with the constant expected count of rarely visited sites. The authors then establish sharp tail bounds $\mathbb{P}(\alpha(n)>a(\log n)^2) \asymp n^{-2a}$ and prove there exists a constant $C\in(1/32,1/2]$ such that $\limsup_{n\to\infty} \frac{\alpha(n)}{(\log n)^2}=C$ almost surely. The approach adapts and extends methods for rarely visited sites, using detailed path decompositions, Catalan-number connections, and independent-block arguments to derive both probabilistic bounds and the almost sure growth law. The results illuminate the fine-scale edge-occupation structure of simple random walks and connect edge-level sparsity to the well-studied behavior of rarely visited sites.

Abstract

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 $α(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(α(n))$, show that $n\to \mathbb{E}(α(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(α(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (α(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{α(n)}{(\log n)^2}=C$ almost surely.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• proof : Proof of Theorem \ref{['thm: expectation of alpha(n)']}
• Remark 2.1
• Lemma 3.1
• Lemma 3.2
• proof : Proof.
• proof : Proof of Lemma \ref{['lem: main lem']}.