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

Recently, an analytic method was developed to study in the large N limit no n-Hermitian random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the exist ing Gaussian non-Hermitian literature. One obtains an explicit algebraic eq uation for the integrated density of eigenvalues from which the Green's fun ction and averaged density of eigenvalues could be calculated in a simple m anner. Thus, that formalism may be thought of as the non-Hermitian analog o f the method due to Brezin, Itzykson, Parisi, and Zuber for analyzing Hermi tian non-Gaussian random matrices. A somewhat surprising result is the so c alled "single ring" theorem, namely, that the domain of the eigenvalue dist ribution in the complex plane is either a disk or an annulus. In this artic le we extend previous results and provide simple new explicit expressions f or the radii of the eigenvalue distribution and for the value of the eigenv alue density at the edges of the eigenvalue distribution of the non-Hermiti an matrix in terms of moments of the eigenvalue distribution of the associa ted Hermitian matrix. We then present several numerical verifications of th e previously obtained analytic results for the quartic ensemble and its pha se transition from a disk shaped eigenvalue distribution to an annular dist ribution. 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. (C) 2001 American Insti tute of Physics.