Mixing time for random walk on supercritical dynamical percolation
成果类型：
Article
署名作者：
Peres, Yuval; Sousi, Perla; Steif, Jeffrey E.
署名单位：
Microsoft; University of Cambridge; Chalmers University of Technology; University of Gothenburg
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-019-00927-z
发表日期：
2020
页码：
809-849
关键词：
摘要：
We consider dynamical percolation on the d-dimensional discrete torus Zndof side length n, where each edge refreshes its status at rate mu=mu n <= 1/2 to be open with probability p. We study random walk on the torus, where the walker moves at rate 1 / (2d) along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case p1/2. When theta(p)>0, we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are based on percolation results (e.g., the Grimmett-Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time.
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