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Using the theory of negative association for measures and the notion ofrandom weak limits of sparse graphs, we establish the validity of the cavitymethod for counting spanning subgraphs subject to local constraints inasymptotically tree-like graphs. Specifically, the corresponding free entropydensity is shown to converge along any sequence of graphs whose random weaklimit is a tree, and the limit is directly expressed in terms of the uniquesolution to a limiting cavity equation. On a Galton-Watson tree, the lattersimplifies into a recursive distributional equation which can be solvedexplicitely. As an illustration, we provide an explicit-limit formula for the$b-$matching number of an Erd\H{o}s-R\'enyi random graph with fixed averagedegree and diverging size, for any $b\in\mathbb N$.

The cavity method for counting spanning subgraphs subject to local constraints