Fast Sampling of $b$-Matchings and $b$-Edge Covers

Abstract: For an integer $b \ge 1$, a $b$-matching (resp. $b$-edge cover) of a graph $G=(V,E)$ is a subset $S\subseteq E$ of edges such that every vertex is incident with at most (resp. at least) $b$ edges from $S$. We prove that for any $b \ge 1$ the simple Glauber dynamics for sampling (weighted) $b$-matchings and $b$-edge covers mixes in $O(n\log n)$ time on all $n$-vertex bounded-degree graphs. This significantly improves upon previous results which have worse running time and only work for $b$-matchings with $b \le 7$ and for $b$-edge covers with $b \le 2$. More generally, we prove spectral independence for a broad class of binary symmetric Holant problems with log-concave signatures, including $b$-matchings, $b$-edge covers, and antiferromagnetic $2$-spin edge models. We hence deduce optimal mixing time of the Glauber dynamics from spectral independence. The core of our proof is a recursive coupling inspired by (Chen and Zhang '23) which upper bounds the Wasserstein $W_1$ distance between distributions under different pinnings. Using a similar method, we also obtain the optimal $O(n\log n)$ mixing time of the Glauber dynamics for the hardcore model on $n$-vertex bounded-degree claw-free graphs, for any fugacity $\lambda$. This improves over previous works which have at least cubic dependence on $n$.