A tomography of the GREM: Beyond the REM conjecture

In a companion paper we proved that in a large class of Gaussian disordered spin systems the local statistics of energy values near levels N1/2+alpha with alpha < 1/2 are described by a simple Poisson process. In this paper we address the issue as to whether this is optimal, and what will happen if alpha=1/2. We do this by analysing completely the Gaussian Generalised Random Energy Models (GREM). We show that the REM behaviour persists up to the level beta N-c, where beta(c) denotes the critical temperature. We show that, beyond this value, the simple Poisson process must be replaced by more and more complex mixed Poisson point processes.