Exact Limsup Growth of Rarely Visited Sites for One-Dimensional Simple Random Walk

Abstract

We investigate the minimal local time $f(n)$ of a one-dimensional simple random walk up to time $n$, defined as the smallest number of visits to any site in the range. A conjecture formulated repeatedly by Erd\H{o}s and R\'{e}v\'{e}sz (1987, 1991) stated that $\limsup_{n\to\infty}f(n)=2$ almost surely, which was disproved by T\'{o}th (1996) who showed $\limsup_{n\to\infty}f(n)=\infty$. Subsequently, R\'{e}v\'{e}sz (2013) suggested studying the growth rate and established an upper bound of the order $\log n$. In this paper, we determine the precise asymptotic growth rate, proving that with probability one, $$ \limsup_{n\to\infty}\frac{f(n)}{\log\log n}=\frac{1}{\log 2}. $$ This result answers the open question posed in Section 13.2 of R\'{e}v\'{e}sz (2013).