Statistical properties of many-particle spectra. II. Two-point correlations and fluctuations

The two-point correlation function for complex spectra described by the Gaussian Orthogonal Ensemble (GOE) is calculated, and its essential simplicity displayed, by an elementary procedure which derives from orthogonal invariance and the dominance of intrinsic binary correlations. The resultant function is used for an approximate calculation of the standard fluctuation measures. Good agreements are found with exact results where these are available, this incidentally demonstrating that the measures are, for the most part, two-point measures. It is shown that they vary slowly over the spectrum, a result which is in agreement both with experiment and with Monte Carlo calculations. The same technique can be used for higher-order correlation functions, and possibly also for more complicated ensembles in which case the results would be relevant to the question why GOE fluctuations give a good account of experimental results.