The supercritical regime in the normal matrix model with cubic potential

TL;DR

This paper studies the supercritical regime of the normal matrix model with cubic potential, revealing new spectral curve behaviors, critical phenomena, and asymptotics of orthogonal polynomials using Riemann-Hilbert analysis.

Contribution

It extends the analysis of the normal matrix model to the supercritical regime, identifying spectral curve evolution and new critical behaviors.

Findings

Spectral curve satisfies Boutroux condition in supercritical regime

Existence of a second critical point where the motherbody shrinks to the origin

Strong asymptotics of orthogonal polynomials show zeros tend to the spectral contours

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 $\Sigma_1$, 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 $3 \times 3$ matrix valued Riemann-Hilbert problem. It follows that the zeros of the orthogonal polynomials tend to $\Sigma_1$, with the exception of at most three spurious zeros.