Multiple orthogonal polynomials, string equations and the large-n limit

TL;DR

This paper develops a method to analyze the large-n limit of multiple orthogonal polynomials using Riemann-Hilbert problems, string equations, and integrable systems, with applications to random matrices and Brownian motions.

Contribution

It introduces a novel approach combining Riemann-Hilbert problems and string equations to study the asymptotic behavior of multiple orthogonal polynomials.

Findings

Derived string equations from Riemann-Hilbert problems for multiple orthogonal polynomials.

Provided a method to determine the quasiclassical limit in the phase space of the Whitham hierarchy.

Applied the framework to analyze large-n limits in random matrix ensembles and non-intersecting Brownian motions.

Abstract

The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated to pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the phase space of the Whitham hierarchy of dispersionless integrable systems is provided. Applications to the analysis of the large-n limit of multiple orthogonal polynomials and their associated random matrix ensembles and models of non-intersecting Brownian motions are given.