  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 $n$.
