TY - JOUR

T1 - Asymptotics for the Partition Function in Two-Cut Random Matrix Models

AU - Claeys, T.

AU - Grava, T.

AU - McLaughlin, K. D.T.R.

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/10/25

Y1 - 2015/10/25

N2 - 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(xi-xj)2∏j=1Ne-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 ZN(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 ZN(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 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 to this point only been successful to compute asymptotics for the partition function in the one-cut case.

AB - 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(xi-xj)2∏j=1Ne-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 ZN(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 ZN(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 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 to this point only been successful to compute asymptotics for the partition function in the one-cut case.

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U2 - 10.1007/s00220-015-2412-y

DO - 10.1007/s00220-015-2412-y

M3 - Article

AN - SCOPUS:84937920523

SN - 0010-3616

VL - 339

SP - 513

EP - 587

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -