Cutoff for permuted Markov chains

IF 1.5 Q2 PHYSICS, MATHEMATICAL
Anna Ben-Hamou, Y. Peres
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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).
置换马尔可夫链的截止
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。
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来源期刊
CiteScore
2.30
自引率
0.00%
发文量
16
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