置换马尔可夫链的截止

IF 1.5 Q2 PHYSICS, MATHEMATICAL
Anna Ben-Hamou, Y. Peres
{"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":null,"pages":null},"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\":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}","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

摘要

设$P$为大小为$n$的双随机矩阵,设$\Pi$为大小为$n$的置换矩阵。本文主要研究转移矩阵由$Q=P\Pi$给出的马尔可夫链的混合时间。换句话说,这个链在由$P$控制的随机步骤和由$\Pi$控制的确定性步骤之间交替。我们证明,如果排列$\Pi$是均匀随机选择的,那么在$P$的温和假设下,链$Q$有高概率在$\frac{\log n}{\mathbf{h}}$时间表现出截断,其中$\mathbf{h}$是$P$的熵率。此外,对于确定性排列,我们改进了Chatterjee和Diaconis(2020)获得的混合时间上界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Cutoff for permuted Markov chains
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).
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信