Abstract |
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We obtain large N asymptotics for the Hermitian random matrix partition function ZN(V)=∫RN∏i<j(xi−xj)2∏j=1Ne−NV(xj)dxj, 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 logZN(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 logZN(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 logZN(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. |