{"title":"Cutoff for permuted Markov chains","authors":"Anna Ben-Hamou, Y. Peres","doi":"10.1214/22-aihp1248","DOIUrl":null,"url":null,"abstract":"Let $P$ be a bistochastic matrix of size $n$, and let $\\Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=P\\Pi$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $\\Pi$. We show that if the permutation $\\Pi$ is chosen uniformly at random, then under mild assumptions on $P$, with high probability, the chain $Q$ exhibits cutoff at time $\\frac{\\log n}{\\mathbf{h}}$, where $\\mathbf{h}$ is the entropic rate of $P$. Moreover, for deterministic permutations, we improve the upper bound on the mixing time obtained by Chatterjee and Diaconis (2020).","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1248","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 3
Abstract
Let $P$ be a bistochastic matrix of size $n$, and let $\Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=P\Pi$. In other words, the chain alternates between random steps governed by $P$ and deterministic steps governed by $\Pi$. We show that if the permutation $\Pi$ is chosen uniformly at random, then under mild assumptions on $P$, with high probability, the chain $Q$ exhibits cutoff at time $\frac{\log n}{\mathbf{h}}$, where $\mathbf{h}$ is the entropic rate of $P$. Moreover, for deterministic permutations, we improve the upper bound on the mixing time obtained by Chatterjee and Diaconis (2020).