Statistical properties of many-particle spectra. IV. New ensembles by Stieltjes transform methods

Abstract

New Gaussian matrix ensembles, with arbitrary centroids and variances for the matrix elements, are defined as modifications of the three standard ones—GOE, GUE and GSE. The average density and two-point correlation function are given in the general case in terms of the corresponding Stieltjes transforms, first used by Pastur for the density. It is shown for the centroid-modified ensemble K + αH that when the operator K preserves the underlying symmetries of the standard ensemble H, then, as the magnitude of α grows, the transition of the fluctuations to those of H is very rapid and discontinuous in the limit of asymptotic dimensionality. Corresponding results are found for other ensembles. A similar Dyson result for the effects of the breaking of a model symmetry on the fluctuations is generalized to any model symmetry, as well as to the fundamental symmetries such as time-reversal invariance.