Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales

Motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a GUE. The relative strength of the two contributions is determined by a parameter $\lambda$ suitably defined on the unfolded scale. Using results for the spectral two-point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes. For $\lambda$ << 1 the correlations are given by Dawson's integral while for $\lambda$ >> 1 we derive a novel analytical formula for the two-point function. In both cases the energy scales, in units of the mean level spacing, at which deviations from pure GUE behavior become noticable can be identified. We also derive an exact expansion of the local level density for finite level number.