Mixing time for random walk on supercritical dynamical percolation
Journal article, 2019

We consider dynamical percolation on the d-dimensional discrete torus Znd of side length n, where each edge refreshes its status at rate μ= μn≤ 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 p< pc(Zd) , the (annealed) mixing time of the walk is Θ (n2/ μ) , and it was conjectured that in the supercritical case p> pc(Zd) , the mixing time is Θ (n2+ 1 / μ) ; here the implied constants depend only on d and p. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption θ(p) > 1 / 2. When θ(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.

Dynamical percolation

stopping times.

mixing times

random walk

Author

Yuval Peres

Microsoft Research

Perla Sousi

University of Cambridge

Jeffrey Steif

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

Probability Theory and Related Fields

0178-8051 (ISSN) 1432-2064 (eISSN)

Vol. In Press

Stochastics for big data and big systems - bridging local and global

Knut and Alice Wallenberg Foundation, 2013-01-01 -- 2018-09-01.

Färgning av slumpmässiga ekvivalensrelationer, slumpvandringar på dynamisk perkolation och bruskänslighet för gränsgrafen i den Erdös-Renyi-slumpgrafsmodellen

Swedish Research Council (VR), 2017-01-01 -- 2020-12-31.

Roots

Basic sciences

Subject Categories

Probability Theory and Statistics

DOI

10.1007/s00440-019-00927-z

More information

Latest update

12/2/2019