Review of the k-body embedded ensembles of Gaussian random matrices

The embedded ensembles were introduced by Mon and French (1975 Ann. Phys., AT 95 90) as physically more plausible stochastic models of many-body systems governed by one- and two-body interactions than provided by standard random-matrix theory. We review several approaches aimed at determining the spectral density, the spectral fluctuation properties and the ergodic properties of these ensembles: moments methods, numerical simulations, the replica trick, the eigenvector decomposition of the matrix of second moments and supersymmetry, the binary correlation approximation, and the study of correlations between matrix elements.