Rapid mixing of global Markov chains via spectral independence: the unbounded degree case

TL;DR

This paper demonstrates that spectral independence guarantees rapid mixing of various global Markov chains, including Swendsen--Wang and systematic scan dynamics, on graphs with unbounded degree, advancing understanding of correlation decay and convergence rates.

Contribution

It proves that spectral independence ensures polynomial mixing times for key Markov chains on graphs with unbounded degree, extending previous results to more general settings.

Findings

Swendsen--Wang dynamics mix in O((Δ log n)^c) steps under spectral independence.

Systematic scan dynamics have O(Δ^c log n) mixing time for monotone systems.

Established a polynomial dependence of entropy factorization on maximum degree.

Abstract

We consider spin systems on general $n$-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen--Wang dynamics for the $q$-state ferromagnetic Potts model on graphs of maximum degree $\Delta$, where $\Delta$ is allowed to grow with $n$, converges in $O((\Delta \log n)^c)$ steps where $c > 0$ is a constant independent of $\Delta$ and $n$. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is $O(\Delta^c \log n)$ for a constant $c>0$ independent of $\Delta$ and $n$. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. Our result implies optimal $O(\log n)$ mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called $k$-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph.