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

arXiv: https://arxiv.org/abs/2103.07459 · HTML: https://mlsanity.com/paper/2103.07459/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2021.

## Built on by (most cited first, 8 of 8)

- [Learning Hard-Constrained Models with One Sample](https://mlsanity.com/paper/2311.03332/) (2023)
- [Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling](https://mlsanity.com/paper/2207.09102/) (2022)
- [Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition](https://mlsanity.com/paper/2405.19553/) (2024)
- [Fast Doubly-Adaptive MCMC to Estimate the Gibbs Partition Function with Weak Mixing Time Bounds](https://mlsanity.com/paper/2111.07372/) (2021)
- [Estimating Ising Models in Total Variation Distance](https://mlsanity.com/paper/2511.21008/) (2025)
- [Sampling and Identity-Testing Without Approximate Tensorization of Entropy](https://mlsanity.com/paper/2506.23456/) (2025)
- [Wasserstein Contraction of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2605.30253/) (2026)
- [On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy](https://mlsanity.com/paper/2606.12720/) (2026)

## Abstract (arXiv)

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 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 - ε0)Delta for some fixed ε0>0. We also obtain O(logn) 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.