The dbar steepest descent method for orthogonal polynomials on the real line with varying weights

TL;DR

This paper develops a new asymptotic analysis technique for orthogonal polynomials with varying weights, extending universality results in random matrix theory by analyzing Riemann-Hilbert problems with nonanalytic jumps.

Contribution

It introduces a novel method for asymptotic analysis of matrix Riemann-Hilbert problems applicable near transition points, broadening the scope of universality in eigenvalue statistics.

Findings

Derived Plancherel-Rotach asymptotics in all complex regions

Extended universality class for eigenvalue statistics

Developed new technique for nonanalytic jump matrices in Riemann-Hilbert problems

Abstract

We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. Our methodology involves developing a new technique of asymptotic analysis for matrix Riemann-Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.