Communications in Mathematical Physics

, Volume 263, Issue 2, pp 535–552

A Tomography of the GREM: Beyond the REM Conjecture


    • Weierstraß–Institut für Angewandte Analysis und Stochastik
    • Institut für MathematikTechnische Universität Berlin
  • Irina Kurkova
    • Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris 6

DOI: 10.1007/s00220-005-1517-0

Cite this article as:
Bovier, A. & Kurkova, I. Commun. Math. Phys. (2006) 263: 535. doi:10.1007/s00220-005-1517-0


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+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 βcN, where βc 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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© Springer-Verlag Berlin Heidelberg 2006