97-85 Pavel Bleher, Alexander Its
Semiclassical Asymptotics of Orthogonal Polynomials, Riemann-Hilbert Problem, and Universality in the Matrix Model (309K, TeX) Feb 20, 97
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Abstract. We derive semiclassical asymptotics for the orthogonal polynomials on the line with the weight $\exp(-NV(z))$, where $V(z)=\di{tz^2\over 2}+{gz^4\over 4},\;g>0,\;t<0$, is a double-well quartic polynomial. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials. The proof of the asymptotics is based on the analysis of the appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics, we prove the universality of the local distribution of eigenvalues in the matrix model with the double-well quartic interaction in the presence of two cuts.

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