EIGENVALUE CORRELATIONS IN THE CIRCULAR ENSEMBLES

Dyson introduced two types of Brownian-motion ensembles of random matrices for studying approximate symmetries in complex quantum systems. The magnitude or symmetry breaking plays the role of a fictitious time t greater-than-or-equal-to 0. We study here 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 we 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 --> 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. We show elsewhere by a semiclassical calculation for a class of quantum maps with time-reversal breaking that the long-range correlations are identical to those obtained in this paper. Our results thus indicate a universality associated with the 'non-equilibrium' eigenvalue statistics.