Block factorization of the relative entropy via spatial mixing

On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization

For general spin systems, we prove that a contractive coupling for any local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the convergence to stationarity and analytic tools for analyzing the decay of relative entropy. As a corollary of our general results, we obtain $O(n\log{n})$ mixing time and $Ω(1/n)$ modified log-Sobolev constant of …

Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows $O(n\log{n})$ mixing time on any $n$-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity $λ$, we establish $O(n\log{n})$ mixing time for the Glauber dynamics on any $n$-vertex graph of c…

自旋系统中方差与熵分解的组合方法

Combinatorial Approach for Factorization of Variance and Entropy in Spin Systems

提出基于平衡分隔子递归块分解的组合框架，建立自旋系统方差与熵的近似张量化，并推出 Glauber 动力学在树宽图和有界度平面图上的新快速混合界。

Stochastic Darboux transformations for quasi-birth-and-death processes and urn models

We consider stochastic UL and LU block factorizations of the one-step transition probability matrix for a discrete-time quasi-birth-and-death process, namely a stochastic block tridiagonal matrix. The simpler case of random walks with only nearest neighbors transitions gives a unique LU factorization and a one-parameter family of factorizations in the UL case. The block structure considered here yields many more possible factorizations resulting in a much enlarged class of potential applications. By reversing the order of the factors (also known as a Darboux transformation) we get new families…

Dobrushin conditions for systematic scan with block dynamics

We study the mixing time of systematic scan Markov chains on finite spin systems. It is known that, in a single site setting, the mixing time of systematic scan can be bounded in terms of the influences sites have on each other. We generalise this technique for bounding the mixing time of systematic scan to block dynamics, a setting in which a (constant size) set of sites are updated simultaneously. In particular we consider the parameter alpha, corresponding to the maximum influence on any site, and show that if alpha<1 then the corresponding systematic scan Markov chain mixes rapidly. As app…

随机定向块模型中混合三重性

Mixing trichotomy for random walks on directed stochastic block models

本文研究了随机定向块模型中随机游走的混合行为，揭示了亚临界、临界和超临界三种不同混合特性及对应的混合时间行为。