Renormalizing rectangles and other topics in random matrix theory

Abstract

We consider random Hermitian matrices made of complex or real M × N rectangular blocks, where the blocks are drawn from various ensembles. These matrices have N pairs of opposite real nonvanishing eigenvalues, as well as M − N zero eigenvalues (for M > N .) These zero eigenvalues are "kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large N, M limit, in which the "rectangularity" r = M N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large N renormalization techniques. In addition to the kinematical δ-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r − 1| is held fixed as N → ∞, the N non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As r → 1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that r → 1 drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

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

“…(2.10) is an algebraic equation for γ(r) and thus may have several r dependent solutions. In constructing the actual γ(r) one may have to match these solutions smoothly into a single function which increases monotonically from γ(0) = 0 to 2 Of course, F (w) is already known in the literature on chiral and rectangular block random hermitean matrices for the Gaussian distribution [11,15,16,17], as well as for non-Gaussian probability distributions of the form (1.1) with an arbitrary polynomial potential V (φ † φ) [18,19,20].…”

“…As is well known [11,19,20], for V a polynomial of degree p, the Green's function F (w) is given by 11) where…”

“…In [3], two of us proposed a "method of hermitization", whereby a problem involving random non-hermitean matrices can be reduced to a problem involving random hermitean matrices, to which various standard methods (such as the diagrammatic method [7], or the "renormalization group" method [8,9,10,11]) can be applied. An idea similar to the "method of hermitization" was expressed independently in [2].…”

Non-hermitian random matrix theory: Method of hermitian reduction

“…The formalism is of course indifferent to the method one may choose to use to determine G µ ν (η; z, z * ). The problem of determining the Green's function of chiral matrices such as H has been discussed by numerous authors [16,20,22,23].…”

“…A renormalization group inspired method used in [13,14,15,16] implicitly involves an expansion in 1/z and is thus also not available for dealing with non-hermitean matrices without suitable further developments. This paper is organized as follows.…”