Semiclassical Asymptotics of Orthogonal Polynomials, Riemann-Hilbert Problem, and Universality in the Matrix Model

Abstract

We derive semiclassical asymptotics for the orthogonal polynomials P n (z) on the line with respect to the exponential weight exp(−N V (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.

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…Such a remarkable connection between the orthogonal polynomials and RH problems is observed by Fokas, Its and Kitaev [16]. Then, we apply the nonlinear steepest descent analysis developed by Deift and Zhou et al [12,13] to the RH problem for Y ; see also Bleher and Its [2]. The idea is to obtain, via a series of invertible transformations Y → T → S → R, eventually the RH problem for R, with jumps close to the identity matrix, where…”

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

“…At the same time, the local statistics of eigenvalues in the large n, N limit satisfies the so-called universality property, i.e. it is determined only by the local characteristics of the eigenvalue density ρ V (compare [9,24,47]). For instance, let us choose a regular point x * ∈ Σ V , i.e.…”

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

“…Here, A (1) , B (1) , A (2) , and B (2) are given by (3.64), (3.65), (3.67), and (3.68), respectively. Now, we are ready to determine the asymptotics of the recurrence coefficients.…”

“…Expanding the jump relation R + = R − v R using (3.54) and (3.60), and collecting the terms with n −1/7 we find R (1) + (z) = R (1)…”