Random walk on dynamical percolation in Euclidean lattices: separating critical and supercritical regimes
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| Format: | Preprint |
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2024
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| _version_ | 1866913568809549824 |
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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 | We study the random walk on dynamical percolation of $\mathbb{Z}^d$ (resp., the two-dimensional triangular lattice $\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $μ\in (0,1/e]$. The random walk moves along open edges in $\mathbb{Z}^d$ (resp., open sites in $\mathcal{T}$) at rate $1$. For the critical regime $p=p_c$, we prove the following two results: on $\mathcal{T}$, the mean squared displacement of the random walk from $0$ to $t$ is at most $O(tμ^{5/132-ε})$ for any $ε>0$; on $\mathbb{Z}^d$ with $d\geq 11$, the corresponding upper bound for the mean squared displacement is $O(t μ^{1/2}\log(1/μ))$. For the supercritical regime $p>p_c$, we prove that the mean squared displacement on $\mathbb{Z}^d$ is at least $ct$ for some $c=c(d)>0$ that does not depend on $μ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15162 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random walk on dynamical percolation in Euclidean lattices: separating critical and supercritical regimes Gu, Chenlin Jiang, Jianping Peres, Yuval Shi, Zhan Wu, Hao Yang, Fan Probability 60K35, 60K37 We study the random walk on dynamical percolation of $\mathbb{Z}^d$ (resp., the two-dimensional triangular lattice $\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $μ\in (0,1/e]$. The random walk moves along open edges in $\mathbb{Z}^d$ (resp., open sites in $\mathcal{T}$) at rate $1$. For the critical regime $p=p_c$, we prove the following two results: on $\mathcal{T}$, the mean squared displacement of the random walk from $0$ to $t$ is at most $O(tμ^{5/132-ε})$ for any $ε>0$; on $\mathbb{Z}^d$ with $d\geq 11$, the corresponding upper bound for the mean squared displacement is $O(t μ^{1/2}\log(1/μ))$. For the supercritical regime $p>p_c$, we prove that the mean squared displacement on $\mathbb{Z}^d$ is at least $ct$ for some $c=c(d)>0$ that does not depend on $μ$. |
| title | Random walk on dynamical percolation in Euclidean lattices: separating critical and supercritical regimes |
| topic | Probability 60K35, 60K37 |
| url | https://arxiv.org/abs/2407.15162 |