DeepAI

# 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(nlogn) mixing time and Ω(1/n) modified log-Sobolev constant of the Glauber dynamics for sampling random q-colorings of an n-vertex graph with constant maximum degree Δ when q > (11/6 - ϵ_0)Δ for some fixed ϵ_0>0. We also obtain O(logn) mixing time and Ω(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.

• 12 publications
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11/04/2020

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

We prove an optimal mixing time bound on the single-site update Markov c...
03/22/2021

### From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk

We show that the existence of a “good” coupling w.r.t. Hamming distance ...
04/17/2018

### Quantum conditional relative entropy and quasi-factorization of the relative entropy

The existence of a positive log-Sobolev constant implies a bound on the ...
03/08/2022

### Localization Schemes: A Framework for Proving Mixing Bounds for Markov Chains

Two recent and seemingly-unrelated techniques for proving mixing bounds ...
11/05/2021

### Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities

We introduce a framework for obtaining tight mixing times for Markov cha...
07/16/2020

### The Swendsen-Wang Dynamics on Trees

The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain...
07/16/2020

### Rapid mixing from spectral independence beyond the Boolean domain

We extend the notion of spectral independence (introduced by Anari, Liu,...