The Exact Limsup Constant for Once-Visited Sites of One-Dimensional Simple Random Walk

TL;DR

The paper determines the exact almost-sure limsup constant for the number of once-visited sites in a one-dimensional simple random walk, showing $\displaystyle \limsup_{n\to\infty} \frac{g_1(n)}{\log^2 n}=\frac{1}{16}$ a.s. The authors introduce a self-boosting iterative framework built on alternating inward/outward excursions and decompose the path into i.i.d. excursions, enabling precise control of rare events where many sites are visited exactly once. Central to the argument are sharp tail bounds for the count of rarely visited sites within conditioned excursion processes and a bootstrapping scheme that ties lower and upper bounds together to pin down the exact constant. The results sharpen the understanding of local times and range growth for 1D SRW and connect to related constants in higher dimensions, illustrating a delicate interplay between excursion structure and extreme-value phenomena. All mathematical notation is presented with explicit $...$ delimitation.

Abstract

For a one-dimensional simple random walk, let $g_1(n)$ denote the number of sites visited exactly once at time $n$. Major (1988) proved that $$\limsup_{n\to\infty}\frac{g_1(n)}{\log^2n}=C\qquad a.s.$$ where $C$ is a positive and finite constant. While this result settled the question of existence, the value of $C$ remained unknown. In this paper, we determine that $C=1/16$. The main novelty of our work lies in introducing a self-boosting iterative framework for analysis.

Theorems & Definitions (25)

• Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof : Proof of Theorem \ref{['main result']} assuming Proposition \ref{['prop 1.3']}
• Proposition 4.1
• Lemma 4.2
• Lemma 4.3