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)=−13M3+tM, where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, t∈C, is partitioned into two phase regions, Oone−cut and Otwo−cut, such that in Oone−cut the equilibrium measure is supported by one Jordan arc (cut) and in Otwo−cut by two cuts. The regions Oone−cut and Otwo−cut are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], 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 (so that the Cauchy–Riemann equations are violated for the endpoints). 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.