 Weighted matching Markov chain

On the matchings in a graph, a lazy Markov chain proposes adding an edge, deleting an edge, or exchanging an edge for the unique matching edge incident with it. A proposal incident with two matching edges is rejected. Select the proposed edge uniformly and use the Metropolis–Hastings acceptance probability for the displayed weights. Symmetric proposals give detailed balance; deletions connect every state to the empty matching in a graph. The normalizing constant is the matching generating function.