Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

TL;DR

This paper explores the two-cut phase region in a complex cubic random matrix ensemble, analyzing the behavior of the endpoints of the cuts and deriving asymptotics for associated orthogonal polynomials.

Contribution

It provides a detailed analysis of the two-cut phase region, proving the analyticity of cut endpoints in the real and imaginary parts of the parameter, and extends the semiclassical asymptotics of orthogonal polynomials.

Findings

Endpoints of cuts are analytic functions of the real and imaginary parts of t.

Derived semiclassical asymptotics of orthogonal polynomials.

Established the structure of phase regions separated by critical curves.

Abstract

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V(M)=-\frac{1}{3}M^3+tM$ where $t$ is a complex parameter. As proven in our previous paper, the whole phase space of the model, $t\in\mathbb C$, is partitioned into two phase regions, $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$, such that in $O_{\mathsf{one-cut}}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{\mathsf{two-cut}}$ by two cuts. The regions $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$ are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In our previous work the one-cut phase region was investigated in detail. In the present paper we investigate the two-cut region. We prove that in the two-cut region the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter $t$, but not of the parameter $t$ itself. We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann--Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of $S$-curves and quadratic differentials.