THE EXCLUSION PROCESS MIXES (ALMOST) FASTER THAN INDEPENDENT PARTICLES

• 成果类型: Article
• 署名作者: Hermon, Jonathan; Pymar, Richard
• 署名单位: University of British Columbia; University of London; Birkbeck University London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1455
• 发表日期: 2020
• 页码: 3077-3123
• 关键词: logarithmic sobolev inequality; spectral-gap; mixing times; cutoff phenomenon; random-walk; profile; bounds

摘要

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 = Omega (log(n/k)n), (2) when gap := the spectral-gap of a single walk is O (1/log(4) n) and k >= n(Omega(1)), (3) when k asymptotic to 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(Omega(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.