Determination of S-curves with applications to the theory of nonhermitian orthogonal polynomials

TL;DR

This paper investigates the determination of S-curves in non-hermitian orthogonal polynomial theory, analyzing phase structures and critical phenomena using complex analysis and numerical methods, with applications to random matrix models.

Contribution

It introduces a combined analytic and numerical approach to determine S-curves and phase structures in non-hermitian orthogonal polynomials, applying it to the cubic model.

Findings

Complete description of phases and critical processes in the cubic model

Analysis of asymptotic zero densities and eigenvalue densities

Method for solving equations for branch points using Abelian integrals

Abstract

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.