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

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 \begin{equation*} \limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=C\qquad a.s. \end{equation*} where $C$ is a positive and finite constant. While this result settled the question of existence, the exact 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.