RENORMALIZING RECTANGLES AND OTHER TOPICS IN RANDOM-MATRIX THEORY

We consider random Hermitian matrices made of complex or real M x N re ctangular blocks, where the blocks are drawn from various ensembles. T hese 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 randomne ss. 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 Ga ussian ensembles by a simple diagrammatic method, by the Dyson gas app roach, and by a generalization of the Kazakov method. These methods ma ke use of the invariance of such ensembles under the action of symmetr y groups. The more complicated Wigner ensemble, which does not enjoy s uch symmetry properties, is studied by large-iii renormalization techn iques. In addition to the kinematical delta-function spike in the eige nvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if \r - 1\ is held fixed as N --> infinity, t he N nonzero eigenvalues give rise to two separated lobes that are loc ated symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue d istribution near the endpoints of the lobes, a behavior governed by Ai ry functions. As r --> 1 the lobes come closer, and the Airy oscillato ry behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that r --> 1 drives a crossover t o the oscillation governed by Bessel functions near the origin for mat rices made of square blocks.