Annealing ratios for a matching generating function ($Z_G(b)/Z_G(a)={\mathbb{E}}_{{\pi }_a}(b/a{)}^{|M|}$)

For $0<a\le b$, expand the expectation under the weighted matching Markov chain stationary weights to obtain the displayed identity. If the largest matching in a graph has $n$ edges and $b/a\le 1+1/n$, the observable lies in $[1,e]$. Its sample mean has controlled error by the Hoeffding inequality. A product of $J$ ratios, each estimated to relative accuracy $O(\epsilon /J)$, approximates the full ratio to relative accuracy $O(\epsilon )$.