A characterization of L2 mixing and hypercontractivity via hitting times and maximal inequalities

• 成果类型: Article
• 署名作者: Hermon, Jonathan; Peres, Yuval
• 署名单位: University of California System; University of California Berkeley; Microsoft
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0769-x
• 发表日期: 2018
• 页码: 769-800
• 关键词: logarithmic sobolev inequalities

摘要

There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the L-2 mixing time, tau(2) (while there are sophisticated analytic tools to bound tau(2), in general they do not determine tau(2) up to a constant factor and they lack a probabilistic interpretation). In this work we show that tau(2) can be characterized up to a constant factor using hitting times distributions. We also derive a newextremal characterization of the Log-Sobolev constant, c(LS), as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of cLS in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, tau(2) is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, tau(2) is robust under bounded perturbations of the edge weights.