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

We investigate the large N limit of spectral measures of matrices which relate to the Gibbs measures of a number of statistical mechanical systems on random graphs. These include the Ising and Potts models on random graphs. 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).