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

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, $O_{\mathsf{one}-\mathsf{cut}}$ and $O_{\mathsf{two}-\mathsf{cut}}$, such that in $O_{\mathsf{one}-\mathsf{cut}}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{\mathsf{two}-\mathsf{cut}}$ by two cuts. The regions $O_{\mathsf{one}-\mathsf{cut}}$ and $O_{\mathsf{two}-\mathsf{cut}}$ 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.

References in corpus (1)

• On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight