Paper Info
Title
On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization
Abstract
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 $\Omega(1/n)$ modified log-Sobolev constant of the Glauber dynamics for sampling random $q$-colorings of an $n$-vertex graph with constant maximum degree $\Delta$ when $q > (11/6 - \epsilon_0)\Delta$ for some fixed $\epsilon_0>0$. We also obtain $O(\log{n})$ mixing time and $\Omega(1)$ modified log-Sobolev constant of the Swendsen-Wang dynamics for the ferromagnetic Ising model on an $n$-vertex graph of constant maximum degree when the parameters of the system lie in the tree uniqueness region. At the heart of our results are new techniques for establishing spectral independence of the spin system and block factorization of the relative entropy. On one hand we prove that a contractive coupling of a local Markov chain implies spectral independence of the Gibbs distribution. On the other hand we show that spectral independence implies factorization of entropy for arbitrary blocks, establishing optimal bounds on the modified log-Sobolev constant of the corresponding block dynamics.
Year
DOI
Venue
2022
10.1137/1.9781611977073.145
ACM-SIAM Symposium on Discrete Algorithms (SODA)
DocType
Citations 
PageRank 
Conference
0
0.34
References 
Authors
0
6
Name
Order
Citations
PageRank
Antonio Blanca1149.74
Pietro Caputo200.34
Zongchen Chen326.13
Daniel Parisi400.68
Daniel Štefankovič501.01
Eric Vigoda674776.55