"Single ring theorem" and the disk-annulus phase transition

Recently, an analytic method was developed to study in the large N limit non-Hermitian random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the existing Gaussian non-Hermitian literature. One obtains an explicit algebraic equation for the integrated density of eigenvalues from which the Green's function and averaged density of eigenvalues could be calculated in a simple manner. Thus, that formalism may be thought of as the non-Hermitian analog of the method due to Brézin, Itzykson, Parisi, and Zuber for analyzing Hermitian non-Gaussian random matrices. A somewhat surprising result is the so called "single ring" theorem, namely, that the domain of the eigenvalue distribution in the complex plane is either a disk or an annulus. In this article we extend previous results and provide simple new explicit expressions for the radii of the eigenvalue distribution and for the value of the eigenvalue density at the edges of the eigenvalue distribution of the non-Hermitian matrix in terms of moments of the eigenvalue distribution of the associated Hermitian matrix. We then present several numerical verifications of the previously obtained analytic results for the quartic ensemble and its phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we demonstrate numerically the "single ring" theorem for the sextic potential, namely, the potential of lowest degree for which the "single ring" theorem has nontrivial consequences.