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

Mateusz Piorkowski

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.

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 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.

Paper Structure

This paper contains 15 sections, 7 theorems, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Outline of this paper
  4. Approximating solutions of R-H problems
  5. Two R-H problems
  6. Singular integral formulation of R-H problems
  7. Residual R-H problem
  8. Riemann--Hilbert problems in applications
  9. Application to Orthogonal Polynomials on
  10. Riemann--Hilbert formulation of orthogonal polynomials
  11. Large limit without a parametrix solution
  12. R-H problem with constant jump matrices
  13. Discussion
  14. Airy R-H problem
  15. Local a priori -estimate

Key Result

Proposition 2.1

Let $(v_\mathcal{R},\Sigma)$ be the data of a R-H problem, and assume that $w_\mathcal{R} := v_\mathcal{R}-\mathbb{I} \in L^p(\Sigma)$. Then there is a bijection between R-H solutions $R$, satisfying and solutions $\Phi \in M_{w_\mathcal{R}}^\Sigma$ of Moreover the relation between $R$ and $\Phi$ is given by

Figures (5)

  • Figure 1: Neighbourhood $\mathcal{L}$.
  • Figure 2: Jump contour for $S$.
  • Figure 3: Contour for the Bessel R-H problem
  • Figure 4: Contour for the Airy parametrix problem
  • Figure 5: Contour for the KdV equation with steplike initial data

Theorems & Definitions (16)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Theorem 4.1
  • Theorem 4.2
  • ...and 6 more