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

We obtain large N asymptotics for the random matrix partition function$$Z_N(V)=\int_{\mathbb{R}^N} \prod_{i < j}(x_i-x_j)^2\prod_{j=1}^Ne^{-NV(x_j)}dx_j,$$ZN(V)=∫^RN∏i<j^(xi-xj)2∏j=1N^e-NV(xj)dxj,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 $${N \to \infty}$$N→∞. 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 $${N \to \infty}$$N→∞ 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.