“Single ring theorem” and the disk-annulus phase transition

Abstract

Recently, an analytic method was developed to study in the large N limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the existing Gaussian non-hermitean 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-hermitean analog of the method due to Brézin, Itzykson, Parisi and Zuber for analyzing hermitean 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 paper we extend previous results and provide simple new explicit expressions for the radii of the eigenvalue distiobution and for the value of the eigenvalue density at the edges of the eigenvalue distribution of the non-hermitean matrix in terms of moments of the eigenvalue distribution of the associated hermitean 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 non-trivial consequences.

Non-Hermitian random matrix theory: summation of planar diagrams, the ‘single-ring’ theorem and the disc–annulus phase transition

“…This section contains a detailed pedagogical exposition of the diagrammatic method and its large-N behavior, which is the basis for the discussion in that section. In Section 4 I present numerical simulations of the quartic ensemble, carried in [3], and demonstrate that they fit well with the theoretical predictions of the formalism developed in Section 3. In Section 5 I formulate and prove the "Single Ring" Theorem of [2], according to which, in the large-N limit, the shape of the eigenvalue distribution associated with any of the ensembles studied in Section 3 is either a disk or an annulus.…”

“…It can be shown [3], however, that when a → 0, that is, in the annular to disk transition, < 1/σ > remains finite, but < 1/σ 2 > diverges like 1/ √ a. Thus, from (78) we see that R inner (a = 0), the critical inner radius, is finite.…”

“…(a)-(c) we display our numerical results for ρ annulus (r) for matrices of various sizes, and compare them to the analytical large-N result (5.18) of [2] (or (4.13) of [3]). In these figures we hold −m 2 = µ 2 = 0.5 fixed, and increase g from 0.025 to 0.1.…”

“…I review aspects of work done in collaboration with A. Zee and R. Scalettar [1,2,3] on complex non-hermitean random matrices. I open by explaining why the bag of tools used regularly in analyzing hermitean random matrices cannot be applied directly to analyze non-hermitean matrices, and then introduce the Method of Hermitization, which solves this problem.…”

Random matrices close to Hermitian or unitary: overview of methods and results

“…We do not discuss at all interesting questions related to strongly non-Hermitian matrices, see e.g. papers [48] and references therein.The first weakly non-Hermitian ensemble analysed in full generality [50] was that of non-Hermitian Gaussian deformations, with the anti-Hermitian part taken independently from the GUE. For this case one can develop a rigorous mathematical theory based on the method of orthogonal polynomials.…”