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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.15079 |
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| _version_ | 1866916331483299840 |
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| author | Gu, Chenlin Jiang, Jianping Peres, Yuval Shi, Zhan Wu, Hao Yang, Fan |
| author_facet | Gu, Chenlin Jiang, Jianping Peres, Yuval Shi, Zhan Wu, Hao Yang, Fan |
| contents | Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $μ>0$, and following the refresh, each edge is open independently with probability $p$. The random walk traverses $G$ only along open edges, moving at rate $1$. In the critical regime $p=p_c$, we prove that the speed of the random walk is at most $O(\sqrt{μ\log(1/μ)})$, provided that $μ\le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $μ)$, while in the subcritical regime $p<p_c$, the speed is of order $μ\wedge 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Speed of random walk on dynamical percolation in nonamenable transitive graphs Gu, Chenlin Jiang, Jianping Peres, Yuval Shi, Zhan Wu, Hao Yang, Fan Probability Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $μ>0$, and following the refresh, each edge is open independently with probability $p$. The random walk traverses $G$ only along open edges, moving at rate $1$. In the critical regime $p=p_c$, we prove that the speed of the random walk is at most $O(\sqrt{μ\log(1/μ)})$, provided that $μ\le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $μ)$, while in the subcritical regime $p<p_c$, the speed is of order $μ\wedge 1$. |
| title | Speed of random walk on dynamical percolation in nonamenable transitive graphs |
| topic | Probability |
| url | https://arxiv.org/abs/2407.15079 |