Statistical properties of many-particle spectra: III. Ergodic behavior in random-matrix ensembles

The ergodic problem is defined for random-matrix ensembles and some conditions for ergodicity given. Ergodic properties are demonstrated for the orthogonal, unitary and symplectic cases of the Gaussian and circular ensembles, and also for the Poisson ensemble. The one-point measures, viz., the eigenvalue density, the number statistic and the k'thnearest-neighbor spacings are shown to be ergodic and the ensemble variances of the corresponding spectral averages are explicitly calculated. It is moreover shown, by using Dyson's cluster functions, that all the k-point correlation functions are themselves ergodic as are therefore the fluctuation measures which follow from them. It is proved also that the local fluctuation properties of the Gaussian ensembles are stationary over the spectrum.