Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two--matrix model

TL;DR

This paper develops strong asymptotic formulas for Cauchy biorthogonal polynomials using advanced Riemann-Hilbert analysis, applicable to complex two-matrix models with multiple intervals and arbitrary genus.

Contribution

It introduces a novel Riemann-Hilbert approach for asymptotics of biorthogonal polynomials in complex multi-interval settings, extending previous methods.

Findings

Derived strong asymptotics for Cauchy biorthogonal polynomials.

Solved the Riemann-Hilbert problem using spinorial line bundles.

Applicable to two-matrix models with multiple intervals and arbitrary genus.

Abstract

We apply the nonlinear steepest descent method to a class of 3x3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.