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

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.

References in corpus (4)

• Phase Structure of a Brane/Anti-Brane System at Large N
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies
• An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model

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• Critical measures for vector energy: global structure of trajectories of quadratic differentials
• Multiple phases and meromorphic deformations of unitary matrix models
• Investigation of the two-cut phase region in the complex cubic ensemble of random matrices
• Gravitational lensing by eigenvalue distributions of random matrix models