, Volume 263, Issue 2, pp 535-552
Date: 23 Feb 2006

A Tomography of the GREM: Beyond the REM Conjecture

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Abstract

In a companion paper we proved that in a large class of Gaussian disordered spin systems the local statistics of energy values near levels N 1/2+ α with α<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 α=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 β c N, where β 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.

Communicated by M. Aizenman
Research supported in part by the DFG in the Dutch-German Bilateral Research Group ``Mathematics of Random Spatial Models from Physics and Biology'' and by the European Science Foundation in the Programme RDSES.