Eigenvalue correlations in the circular ensembles

Dyson (1972) introduced two types of Brownian-motion ensembles of random matrices for studying approximate symmetries in complex quantum systems. The magnitude of symmetry breaking plays the role of a fictitious time t>or=0. The authors study the eigenvalue correlations in the circular-type ensembles which serve as models for the evolution operators of quantum maps with chaotic classical limits. In two cases involving time-reversal symmetry breaking they evaluate explicitly the eigenangle-density correlation functions of all orders for all t and for all values of the matrix dimensionality N. The general case is described by a hierarchic set of relations among the correlation functions. As a function of t, the transition in the correlations is found to be rapid for large N, discontinuous for N to infinity . As a function of a local parameter Lambda , which measures the mean square symmetry-admixing matrix element in units of the local average spacing, the transition is found to be smooth. The same Lambda -dependent results were found earlier for the Gaussian-type ensembles which serve as models for the Hamiltonian operators of autonomous chaotic systems.