Asymptotics for the Partition Function in Two-Cut Random Matrix Models

We obtain large N asymptotics for the random matrix partition function Z(N)(V) = integral(RN) Pi(i<j) (x(i) - x(j))(2) Pi(N)(j=1) e(-NV(xj))dx(j), in the case where V is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals (the two-cut case). We compute leading and sub-leading terms in the asymptotic expansion for log Z (N) (V), up to terms that are small as . Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential V. Afterwards, we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two-cut potentials V. The asymptotic expansion of log Z (N) (V) as contains terms that depend analytically on the potential V and that have already appeared in the literature. In addition, our method allows us to compute the V-independent terms of the asymptotic expansion of log Z (N) (V) which, to the best of our knowledge, had not appeared before in the literature. We use rigorous orthogonal polynomial and Riemann-Hilbert techniques, which had to this point only been successful to compute asymptotics for the partition function in the one-cut case.