The supercritical regime in the normal matrix model with cubic potential

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.