Asymptotics for the partition function in two-cut random matrix models

Abstract

We obtain large N asymptotics for the Hermitian random matrix partition function \[Z_N(V)=\int_{\mathbb R^N}\prod_{i<j}(x_i-x_j)^2 \prod_{j=1}^N e^{-N V(x_j)}dx_j,\] in the case where the external potential $V$ is a polynomials 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 $N$ goes to infinity. 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 $N$ goes to infinity contains terms that depend analytically on the potential $V$ and that have already appeared in the literature. In addition our method allows 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 so far been successful to compute asymptotics for the partition function only in the one-cut case.