Matchings on infinite graphs

We consider the Boltzmann distribution with positive fugacity over the matchings of a finite graph, and we establish its weak convergence as the underlying graph converges locally to an infinite graph with finite branching number. As a by-product, we obtain convergence of the scaled logarithm of the matching polynomial. By letting then the fugacity tend to infinity, we obtain a limit theorem for the asymptotic size of a maximum matching in the graph sequence. joint work with Charles Bordenave and Justin Salez.