{"title":"置换马尔可夫链的截止","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":"84 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"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\":\"84 1\",\"pages\":\"\"},\"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}","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}
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).