Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials

TL;DR

This paper proposes a method to derive asymptotic results for orthogonal polynomials without explicitly solving local parametrix problems in the Riemann-Hilbert framework, relying instead on a priori estimates.

Contribution

It introduces a new approach to Riemann-Hilbert problems that bypasses local parametrix solutions, expanding applicability to more irregular weight functions.

Findings

Achieved asymptotic results for orthogonal polynomials with complex weight functions.

Demonstrated the method's potential for implications in random matrix theory.

Provided a framework for handling weights too irregular for traditional local parametrix methods.

Abstract

We study whether in the setting of the Deift-Zhou nonlinear steepest descent method one can avoid solving local parametrix problems explicitly, while still obtaining asymptotic results. We show that this can be done, provided an a priori estimate for the exact solution of the Riemann-Hilbert problem is known. This enables us to derive asymptotic results for orthogonal polynomials on $[-1,1]$ with a new class of weight functions. In these cases, the weight functions are too badly behaved to allow a reformulation of a local parametrix problem to a global one with constant jump matrices. Possible implications for edge universality in random matrix theory are also discussed.