Abstract
Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. We review some rigorous results confirming the validity of this conjecture. In the context of the SK models, we analyse the limits of the validity of the conjecture for energy levels growing with the volume of the system. In the case of the Generalised Random energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies E N < β c N, where β c is the critical temperature. We also explain the more complex behaviour that sets in at higher energies.
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References
H. Bauke and St. Mertens, Universality in the level statistics of disordered systems. Phys. Rev. E, 70:025102(R) (2004).
St. Mertens, Phase transition in the number partitioning problem, Phys. Rev. Letts. 81:4281–4284 (1998).
St. Mertens, Random Costs in Combinatorial Optimization. Phys. Rev. Letts. 84:1347–1350 (2000).
St. Mertens, A physicist’s approach to number partitioning. Phase transitions in combinatorial problems (Trieste, 1999). Theoret. Comput. Sci. 265:79–108 (2001).
C. Borgs, J. Chayes and B. Pittel, Phase transition and finite-size scaling for the integer partitioning problem. Random Struct. Algorithms 19:247–288 (2001).
M. R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer-Verlag, New York (1983).
A. Bovier and I. Kurkova, Poisson convergence in the restricted k-partitioning problem. WIAS preprint 964, to appear in Random Structures & Algorithms cond-mat/0409532.
A. Bovier and I. Kurkova, Local energy statistics in disordered system: a proof of the local REM conjecture. Commun. Math. Phys. 263:513–533 (2006).
C. Borgs, J. T. Chayes, S. Mertens and B. Pittel, Phase diagram for the constrained integer partitioning problem. Random Struct. Algorithms 24:315–380 (2004).
C. Borgs, J. T. Chayes, S. Mertens and Ch. Nair, Proof of the local REM conjecture for number partitioning I: Constant energy scales, http://arxiv.org/abs/cond-mat/0501760, to appear in Random Struct, Algorithms.
C. Borgs, J. T. Chayes, S. Mertens and Ch. Nair, Proof of the local REM conjecture for number partitioning II: Growing energy scales. http://arxiv.org/abs/cond-mat/0508600
A. Bovier, Statistical mechanics of disordered systems. Cambridge University Press (2006).
A. Kuptsov, private communication. To be detailed in G. Ben Arous, V. Gayrard, and A. Kuptsov, A new variant of the REM universality, to appear.
H. Koch and J. Piasko, Some rigorous results on the Hopfield neural network model. J. Stat. Phys. 55:903–928 (1989).
A. Bovier and I. Kurkova, Derrida’s generalised random energy models. I. Models with finitely many hierarchies, Ann. Inst. H. Poincaré Probab. Statist. 40:439–480 (2004).
A. Bovier and I. Kurkova, A tomography of the GREM: beyond the REM conjecture, Commun. Math. Phys. 263:535–552 (2006).
B. Derrida, Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B 24(3):2613–2626 (1981).
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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. Local Energy Statistics in Spin Glasses. J Stat Phys 126, 933–949 (2007). https://doi.org/10.1007/s10955-006-9141-8
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DOI: https://doi.org/10.1007/s10955-006-9141-8