2014/10/31 by Tom Claeys, Tamara Grava, Kenneth D. T-R McLaughlin · 2 citations
Physics and Astronomy · Mathematics · #math-ph #math.AP #math.MP
paper · pdf · doi:10.1007/s00220-015-2412-y
published as Communications in Mathematical Physics, 339 (2015), no. 2, 513 - 587 · 75 pages. To appear in Comm. Math. Physics
arxiv created 2015/06/25 · arxiv updated 2015/10/07
We obtain large N asymptotics for the Hermitian random matrix partition function ZN(V)=∫\mathbb RN∏i<j(xi-xj)2 ∏j=1N e-N V(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 log ZN(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 ZN(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 ZN(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.