The global parametrix in the Riemann-Hilbert steepest descent analysis for orthogonal polynomials

TL;DR

This paper introduces an alternative method using meromorphic differentials to solve the model Riemann-Hilbert problem in the steepest descent analysis of orthogonal polynomials, replacing the traditional hyperelliptic theta functions.

Contribution

It presents a novel approach employing meromorphic differentials for the model Riemann-Hilbert problem in the multi-cut case, offering an alternative to theta function methods.

Findings

Successful construction of the solution using meromorphic differentials

Simplification of the steepest descent analysis in multi-cut cases

Potential for broader applicability in orthogonal polynomial analysis

Abstract

In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta functions. We present here an alternative approach which uses meromorphic differentials instead of theta functions to construct the solution of the model Riemann-Hilbert problem.