Faster Mixing of the Jerrum-Sinclair Chain
Xiaoyu Chen, Weiming Feng, Zhe Ju, Tianshun Miao, Yitong Yin, Xinyuan Zhang
TL;DR
This work advances the understanding of mixing times for the Jerrum-Sinclair chain in the monomer-dimer model by introducing a local-to-global framework that combines canonical-path ideas with transport-flow techniques from high-dimensional expanders. By establishing local Poincaré and log-Sobolev inequalities via carefully constructed transport flows and then lifting them to global bounds using a concave-Dirichlet-forms condition, the authors obtain a mixing time of T_mix = \widetilde{O}_{\lambda}(Δ^2 m) for the 1/2-lazy JS chain on graphs with maximum degree Δ, improving upon the classical bound for general graphs with unbounded Δ. They also derive a near-linear (in m) bound for Glauber dynamics, T_mix = \widetilde{O}_{\lambda}(Δ^3 m), by a comparison argument, and provide a detailed treatment of the transport-flow construction, moment bounds, and pinning extensions. The approach suggests a broader applicability to localization schemes and other HDX-influenced settings, offering a path toward tighter mixing-time results for a wide class of Markov chains used in sampling combinatorial structures.
Abstract
We show that the Jerrum-Sinclair Markov chain on matchings mixes in time $\widetilde{O}(Δ^2 m)$ on any graph with $n$ vertices, $m$ edges, and maximum degree $Δ$, for any constant edge weight $λ>0$. For general graphs with arbitrary, potentially unbounded $Δ$, this provides the first improvement over the classic $\widetilde{O}(n^2 m)$ mixing time bound of Jerrum and Sinclair (1989) and Sinclair (1992). To achieve this, we develop a general framework for analyzing mixing times, combining ideas from the classic canonical path method with the "local-to-global" approaches recently developed in high-dimensional expanders, introducing key innovations to both techniques.