Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918095976660992 |
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| author | Li, Xinyi Shi, Jialu Xu, Qiheng |
| author_facet | Li, Xinyi Shi, Jialu Xu, Qiheng |
| contents | We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $γ\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for $γ\in (\frac{d+2}{2d},1)$. The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Prévost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$ Li, Xinyi Shi, Jialu Xu, Qiheng Probability 05C81, 60F10, 60G70 We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $γ\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for $γ\in (\frac{d+2}{2d},1)$. The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Prévost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs. |
| title | Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$ |
| topic | Probability 05C81, 60F10, 60G70 |
| url | https://arxiv.org/abs/2411.16398 |