Exact Limsup Growth of Rarely Visited Sites for One-Dimensional Simple Random Walk
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914053838864384 |
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| author | Feng, Chenxu Hao, Chenxu |
| author_facet | Feng, Chenxu Hao, Chenxu |
| contents | 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 $\limsup_{n\to\infty}f(n)=2$ almost surely, which was disproved by Tóth (1996) who showed $\limsup_{n\to\infty}f(n)=\infty$. 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, $$ \limsup_{n\to\infty}\frac{f(n)}{\log\log n}=\frac{1}{\log 2}. $$ This result answers the open question posed in Section 13.2 of Révész (2013). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact Limsup Growth of Rarely Visited Sites for One-Dimensional Simple Random Walk Feng, Chenxu Hao, Chenxu Probability 60F15, 60J55 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 $\limsup_{n\to\infty}f(n)=2$ almost surely, which was disproved by Tóth (1996) who showed $\limsup_{n\to\infty}f(n)=\infty$. 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, $$ \limsup_{n\to\infty}\frac{f(n)}{\log\log n}=\frac{1}{\log 2}. $$ This result answers the open question posed in Section 13.2 of Révész (2013). |
| title | Exact Limsup Growth of Rarely Visited Sites for One-Dimensional Simple Random Walk |
| topic | Probability 60F15, 60J55 |
| url | https://arxiv.org/abs/2509.19809 |