THE INTERCHANGE PROCESS ON HIGH-DIMENSIONAL PRODUCTS

• 成果类型: Article
• 署名作者: Hermon, Jonathan; Salez, Justin
• 署名单位: University of British Columbia; Universite PSL; Universite Paris-Dauphine; Universite PSL
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1583
• 发表日期: 2021
• 页码: 84-98
• 关键词: logarithmic sobolev inequality; mixing times; cutoff; cycles

摘要

We resolve a long-standing conjecture of Wilson (Ann. Appl. Probab. 14 (2004) 274-325), reiterated by Oliveira (2016), asserting that the mixing time of the interchange process with unit edge rates on the n-dimensional hyper-cube is of order n. This follows from a sharp inequality established at the level of Dirichlet forms, from which we also deduce that macroscopic cycles emerge in constant time, and that the log-Sobolev constant of the exclusion process is of order 1. Beyond the hypercube, our results apply to cartesian products of arbitrary graphs of fixed size, shedding light on a broad conjecture of Oliveira