First order asymptotics of matrix integrals ; a rigorous approach towards the understanding of matrix models

Abstract

We investigate the limit behaviour of the spectral measures of matrices following the Gibbs measure for the Ising model on random graphs, Potts model on random graphs, matrices coupled in a chain model or induced QCD model. For most of these models, we prove that the spectral measures converge almost surely and describe their limit via solutions to an Euler equation for isentropic flow with negative pressure $p(\rho)=-3^{-1}\pi^2 \rho^3$.