{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:R7N23PBPIIXQYX63HXYARYFXVF","short_pith_number":"pith:R7N23PBP","schema_version":"1.0","canonical_sha256":"8fdbadbc2f422f0c5fdb3df008e0b7a9548d45a6b81e1f21c2f33f68cde8d50a","source":{"kind":"arxiv","id":"2407.15079","version":1},"attestation_state":"computed","paper":{"title":"Speed of random walk on dynamical percolation in nonamenable transitive graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chenlin Gu, Fan Yang, Hao Wu, Jianping Jiang, Yuval Peres, Zhan Shi","submitted_at":"2024-07-21T07:21:14Z","abstract_excerpt":"Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $\\mu>0$, 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 $O(\\sqrt{\\mu \\log(1/\\mu)})$, provided that $\\mu \\le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $\\mu)$, while in the subcritical regime $p0$, 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 $O(\\sqrt{\\mu \\log(1/\\mu)})$, provided that $\\mu \\le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $\\mu)$, while in the subcritical regime $p