The exclusion process mixes (almost) faster than independent particles

Pymar, Richard and Hermon, J. (2020) The exclusion process mixes (almost) faster than independent particles. Annals of Probability 48 (6), pp. 3077-3123. ISSN 0091-1798.

Abstract

Oliveira conjectured that the order of the mixing time of the exclusion process with k-particles on an arbitrary n-vertex graph is at most that of the mixing-time of k independent particles. We verify this up to a constant factor for d-regular graphs when each edge rings at rate 1/d in various cases: (1) when d = Ω(logn/k n), (2) when gap := the spectral-gap of a single walk is O(1/ log4 n) and k > n Ω(1) , (3) when k ≍ n a for some constant 0 < a < 1. In these cases our analysis yields a probabilistic proof of a weaker version of Aldous’ famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of O(log n log log n/gap), which is within a log log n factor from Oliveira’s conjecture when k > n Ω(1). As applications we get new mixing bounds: (a) O(log n log log n) for expanders, (b) order d log(dk) for the hypercube {0, 1} d , (c) order (Diameter)2 log k for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.

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