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

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 Erdős and Révész (1987, 1991) stated that ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}f(n)=2$ almost surely, which was disproved by Tóth (1996) who showed ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}f(n)=\mathrm{\infty }$. Subsequently, Révész (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, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{f(n)}{\log \log n}=\frac{1}{\log 2}\mathrm{.}$$ This result answers the open question posed in Section 13.2 of Révész (2013).