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

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.

References in corpus (5)

• Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Rarefaction Waves of the Korteweg-de Vries Equation via Nonlinear Steepest Descent
• Long-Time Asymptotics of the Toda Lattice for Decaying Initial Data Revisited
• Shock problem for MKdV equation: Long time Dynamics of the Step-like initial data
• Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight