Favorite sites for simple random walk in two and more dimensions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912702593499136 |
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| author | Hao, Chenxu Li, Xinyi Okada, Izumi Zheng, Yushu |
| author_facet | Hao, Chenxu Li, Xinyi Okada, Izumi Zheng, Yushu |
| contents | On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987), which was brought up again in Dembo (2005). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Favorite sites for simple random walk in two and more dimensions Hao, Chenxu Li, Xinyi Okada, Izumi Zheng, Yushu Probability 60F15, 60J55 On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987), which was brought up again in Dembo (2005). |
| title | Favorite sites for simple random walk in two and more dimensions |
| topic | Probability 60F15, 60J55 |
| url | https://arxiv.org/abs/2409.00995 |