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. Here we give necessary conditions for this hypothesis to be true, which we show to be satisfied in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bauke, H., Franz, S., Mertens, S.: Number partitioning as random energy model. J. Stat. Mech.: Theory and Experiment, page P04003 (2004)
Bauke, H., Mertens, S.: Universality in the level statistics of disordered systems. Phys. Rev. E 70, 025102(R) (2004)
Borgs, C., Chayes, J., Pittel, B.: Phase transition and finite-size scaling for the integer partitioning problem. Random Structures Algorithms 19(3–4), 247–288 (2001)
Borgs, C., Chayes, J.T., Mertens, S., Pittel, B.: Phase diagram for the constrained integer partitioning problem. Random Structures Algorithms 24(3), 315–380 (2004)
Borgs, C., Chayes, J.T., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning. Preprint 2005, available at http://research.microsoft.com/~chayes/
Borgs, C., Chayes, J.T., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning II: Growing energy scales. http://arvix.org/list/ cond-mat/0508600, 2005
Bovier, A.: Statistical mechanics of disordered systems. In: Cambridge Series in Statistical and Probabilisitc mathematics, Cambridge University Press, to appear May 2006
Bovier, A., Kurkova, I.: Derrida's generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40(4), 439–480 (2004)
Bovier, A., Kurkova, I.: Poisson convergence in the restricted k-partioning problem. Preprint 964, WIAS, 2004, available at http://www.wias-berlin.de/people/files/publications.html, to appear in Random Structures Algorithms (2006)
Bovier, A., Kurkova, I.: A tomography of the GREM: beyond the REM conjecture. Commun. Math. Phys. 263(2), 535–552 (2006)
Bovier, A., Kurkov, I., Löwe, M.: Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30, 605–651 (2002)
Bovier, A., Mason, D.: Extreme value behaviour in the Hopfield model. Ann. Appl. Probab. 11, 91–120 (2001)
Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24(5), 2613–2626 (1981)
Derrida, B.: A generalisation of the random energy model that includes correlations between the energies. J. Phys. Lett. 46, 401–407 (1985)
Mertens, S.: Phase transition in the number partitioning problem. Phys. Rev. Lett. 81(20), 4281–4284 (1998)
Mertens, S.: A physicist's approach to number partitioning. Theoret. Comput. Sci. 265(1–2), 79–108, (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
Bovier, A., Kurkova, I. Local Energy Statistics in Disordered Systems: A Proof of the Local REM Conjecture. Commun. Math. Phys. 263, 513–533 (2006). https://doi.org/10.1007/s00220-005-1516-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1516-1