Álvarez Galindo, GabrielMartínez Alonso, LuisMedina Reus, Elena2023-06-192023-06-192013-061742-546810.1088/1742-5468/2013/06/P06006https://hdl.handle.net/20.500.14352/34937©IOP Publishing. We thank Prof. A. Martínez Finkelshtein for useful conversations and for calling our attention to the work [6]. The financial support of the Ministerio de Ciencia e Innovación under projects FIS2008-00200 and FIS2011-22566 is gratefully acknowledged.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.engjournal articlehttp://dx.doi.org/10.1088/1742-5468/2013/06/P06006http://iopscience.iop.orghttp://arxiv.org/abs/1305.3028open access538.9Riemann-hilbert approachValued weight functionMatrix modelsAsymptoticsUniversalityRespectDualityGravityFísica de materialesFísica del estado sólido2211 Física del Estado Sólido