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.
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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.
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Bovier, A., Kurkova, I. A Tomography of the GREM: Beyond the REM Conjecture. Commun. Math. Phys. 263, 535–552 (2006). https://doi.org/10.1007/s00220-005-1517-0
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DOI: https://doi.org/10.1007/s00220-005-1517-0