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 $n$.

翻译：对整数 $b \ge 1$，图 $G=(V,E)$ 的 $b$-匹配（分别地，$b$-边覆盖）是边子集 $S\subseteq E$，使得每个顶点至多（分别地，至少）与 $S$ 中 $b$ 条边关联。我们证明：对任意 $b \ge 1$，用于采样（加权）$b$-匹配和 $b$-边覆盖的简单格劳伯动力学在所有 $n$ 顶点有界度图上以 $O(n\log n)$ 时间混合。这显著改进了先前结果——其运行时间更长且仅适用于 $b \le 7$ 的 $b$-匹配和 $b \le 2$ 的 $b$-边覆盖。更一般地，我们为一类具有对数凹特征函数的二元对称霍尔朗特问题证明了谱独立性，包括 $b$-匹配、$b$-边覆盖和反铁磁 $2$-自旋边模型。因此我们从谱独立性推导出格劳伯动力学的最优混合时间。我们证明的核心是受 (Chen and Zhang '23) 启发的递归耦合，它限定了不同钉扎条件下分布间的瓦瑟斯坦 $W_1$ 距离上界。使用类似方法，我们还获得了任意逸度 $\lambda$ 下，$n$ 顶点有界度无爪图上硬核模型格劳伯动力学的最优 $O(n\log n)$ 混合时间。这改进了先前依赖 $n$ 至少三次方的工作。