Semiclassical asymptotics of orthogonal polynomials , Riemann - Hilbert problem , and universality in the matrix model

We derive semiclassical asymptotics for the orthogonal polynomials P n (z) on the line with respect to the exponential weight exp(−NV (z)), where V (z) is a double-well quartic polynomial, in the limit when n, N → ∞. We assume that ε ≤ (n/N) ≤ λ cr − ε for some ε > 0, where λ cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials, and we show that these coefficients form a cycle of period two which drifts slowly with the change of the ratio n/N. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics of the orthogonal polynomials, 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. Contents 1. Introduction and formulation of the main theorem 2. Universality of the local distribution of eigenvalues in the matrix model 3. The Lax pair for the Freud equation 4. The Stokes phenomenon 5. The Riemann-Hilbert problem 6. Formal asymptotic expansion for R n 7. Proof of the main theorem: Approximate solution to the Riemann-Hilbert problem Appendix A.