Replica variables, loop expansion, and spectral rigidity of random-matrix ensembles

The replica trick of statistical mechanics is used to derive integral representations of n-point Green's functions both for the GOE and the EGOE. These integral representations are particularly suited for perturbative evaluation (loop expansion). Using the one-loop correction to the GOE one-point function, it is found that the density of states at the edge of the semicircle scales is ∼N ^{-1 3}ρ{variant}(N ^{2 3}δ) where N is the dimension of the matrix ensemble. For the n-point functions with n ≥ 2, the existence of the microscopic limit to all orders in N^-1 is proved by decomposing the integration variables into massive (i.e., macroscopic) and massless (microscopic) components. Evaluation of the EGOE two-point function to leading order in the inverse local distance variable yields the first analytic evidence that the long-range correlations of EGOE spectra are similar to the GOE but not-stationary.