### Matchings and rank for random diluted graphs

Lelarge, M *(ENS)*

Thursday 25 March 2010, 14:00-15:00

Seminar Room 1, Newton Institute

#### Abstract

We study matchings on a sequence of random graphs that converge locally to trees. Inspired by techniques from random matrix theory, we rigorously prove the validity of the cavity method for the computation of the entropy. At a positive temperature, the cavity equations are interpreted as equations for the local marginals of the Boltzmann Gibbs distribution in the space of matchings on a (possibly) infinite tree. These equations also appear in the computation of the asymptotic rank of the adjacency matrices of the random graphs. We also define a determinantal process on the tree which is the limit at positive temperature of the matchings on the sequence of graphs. (joint work with Charles Bordenave and Justin Salez)

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