THREE GENERALIZED MATRIX ENSEMBLES: A REVIEW

Abstract

The standard Gaussian matrix ensembles are known for their robustness, in the sense that their level statistics is representative of a large class of matrix models. It is possible, however, to define generalized ensembles which are able to “escape” the Gaussian fixed point and to exhibit statistics radically different from the Wigner-Dyson statistics of the standard ensembles. Such generalized ensembles contain a parameter which is allowed to change with the matrix size N. Depending on the rate of change, one obtains three distinct limiting statistics: the Wigner-Dyson statistics, the Poisson statistics of uncorrelated levels or a new intermediate statistics (the “non-trivial” fixed point). The present review discusses three examples of such generalized ensembles.