Sampling Matchings in Near-linear Time

Abstract: For every fixed activity $λ&gt;0$, we establish three results for the monomer--dimer model on an $n$-vertex simple graph $G$ with $m\ge1$ edges and maximum degree $Δ$. 1. Near-linear mixing and sampling. Single-edge Glauber dynamics has mixing time $O_λ(m[\log² n+\log(1/\varepsilon)])$, giving a near-linear-time approximate sampler. 2. Work-efficient parallel sampling. We simulate the same Glauber dynamics in parallel using $\tilde{O}<em>λ(m+n)$ work and $\tilde{O}</em>λ(\min{Δ,m^{1/3},\sqrt n})$ depth with high probability. 3. Fast approximate counting. We estimate the partition function within relative error $\varepsilon$ in $\tilde{O}_λ(n<sup>2/\varepsilon<sup>2)$ work. For dense graphs with $m=Θ(n<sup>2)$, this is near-linear in the input size. For the mixing theorem, we establish a general log--Sobolev criterion based on field-dynamics spectral stability, with only logarithmic dependence on the inverse occupied-marginal lower bound. Parallelism uses a matching-specific analysis of occupation-interval dependencies. Counting uses monomer-preconditioned Jerrum--Sinclair dynamics, whose parameters are learned efficiently by Glauber dynamics.