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Consider a sequence of edge-weighted random graphs that converges in Benjamini-Schramm sense to a Unimodular Bienaymé Galton Watson tree with i.i.d weights on edges. If the weights are continuous, we prove that the joint distribution of the graphs and any maximum weight matching on the graphs also converges in Benjamini-Schramm sense to an identified joint distribution of the tree and a matching. The limiting joint distribution of the tree and the matching is characterised by the stationary measure of a message-passing algorithm. The proof is built upon Aldous’ original work on the random assignment problem.
We will also explore the case of maximal size maximum weight matchings as an extension and discuss a few open questions and conjectures.
This talk is based on a joint work with Nathanaël Enriquez, Laurent Ménard and Vianney Perchet.
