The Supercritical Regime In The Normal Matrix Model With Cubic Potential

Keywords

Boutroux condition; Multiple orthogonal polynomials; Normal matrix model; Riemann-Hilbert problem; Steepest descent analysis

Abstract

The normal matrix model with a cubic potential is ill-defined and it develops a critical behavior in finite time. We follow the approach of Bleher and Kuijlaars to reformulate the model in terms of orthogonal polynomials with respect to a Hermitian form. This reformulation was shown to capture the essential features of the normal matrix model in the subcritical regime, namely that the zeros of the polynomials tend to a number of segments (the motherbody) inside a domain (the droplet) that attracts the eigenvalues in the normal matrix model.In the present paper we analyze the supercritical regime and we find that the large n behavior is described by the evolution of a spectral curve satisfying the Boutroux condition. The Boutroux condition determines a system of contours σ1, consisting of the motherbody and whiskers sticking out of the domain. We find a second critical behavior at which the original motherbody shrinks to a point at the origin and only the whiskers remain. In the regime before the second criticality we also give strong asymptotics of the orthogonal polynomials by means of a steepest descent analysis of a 3×3 matrix valued Riemann-Hilbert problem. It follows that the zeros of the orthogonal polynomials tend to σ1, with the exception of at most three spurious zeros.

Publication Date

10-1-2015

Publication Title

Advances in Mathematics

Volume

283

Number of Pages

530-587

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.aim.2015.06.020

Socpus ID

84938778578 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84938778578

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