Abstract
We survey in this paper a universality phenomenon which shows that some characteristics of complex random energy landscapes are model-independent, or universal. This universality, called REM-universality, was discovered by S. Mertens and H. Bauke in the context of combinatorial optimization. We survey recent advances on the extent of this REM-universality for equilibrium as well as dynamical properties of spin glasses. We also focus on the limits of REM-universality, i.e., when it ceases to be valid.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bauke, H.: Zur Universalität des Random-Energy-Modells. Ph.D. thesis, University of Magdeburg (2006).
Bauke, H., Franz, S., Mertens, S.: Number partitioning as a random energy model. J. Stat. Mech. Theory Exp. page P04003 (2004).
Bauke, H., Mertens, S.: Universality in the level statistics of disordered systems. Phys. Rev. E (3) 70, 025,102(R) (2004).
Bauke, H., Mertens, S.: Ubiquity of the Random-Energy Model. Poster presented at the “International Summer School: Fundamental Problems in Statistical Physics XI” in Leuven (Belgium) (2005).
Ben Arous, G., Bovier, A., Černý, J.: Universality of random energy model-like ageing in mean field spin glasses. J. Stat. Mech. Theory Exp. (4), L04,003, 8 (2008).
Ben Arous, G., Bovier, A., Černý, J.: Universality of the REM for dynamics of mean-field spin glasses. Comm. Math. Phys. 282(3), 663–695 (2008).
Ben Arous, G., Bovier, A., Gayrard, V.: Aging in the random energy model. Phys. Rev. Lett. 88(8), 87,201-1–87,201-4 (2002).
Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Comm. Math. Phys. 235(3), 379–425 (2003).
Ben Arous, G., Bovier, A., Gayrard, V.: Glauber dynamics of the random energy model. II. Aging below the critical temperature. Comm. Math. Phys. 236(1), 1–54 (2003).
Ben Arous, G., Černý, J.: Dynamics of Trap Models. In: A. Bovier, F. Dunlop, A. Van Enter, F. Den Hollander (eds.) Mathematical Statistical Physics, École d’Été de Physique des Houches: Session LXXXIII: 4–29 July, 2005, pp. 331–394. Elsevier (2006).
Ben Arous, G., Černý, J.: The arcsine law as a universal aging scheme for trap models. Comm. Pure Appl. Math. 61(3), 289–329 (2008).
Ben Arous, G., Gayrard, V., Kuptsov, A.: A new REM conjecture. In: V. Sidoravicius, M.E. Vares (eds.) In and Out of Equilibrium 2, Progress in Probability, vol. 60, pp. 59–96. Birkhäuser (2008).
Ben Arous, G., Kuptsov, A.: The limits of REM Universality, in preparation.
Borgs, C., Chayes, J., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning. I: Constant energy scales. Random Struct. Algorithms 34(2), 217–240 (2009).
Borgs, C., Chayes, J., Mertens, S., Nair, C.: Proof of the local REM conjecture for number partitioning. II. Growing energy scales. Random Struct. Algorithms 34(2), 241–284 (2009).
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). Analysis of algorithms (Krynica Morska, 2000).
Bouchaud, J.P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992).
Bouchaud, J.P., Dean, D.S.: Aging on Parisi’s tree. J. Phys. I (France) 5, 265 (1995).
Bovier, A.: Statistical mechanics of disordered systems, MaPhySto Lecture Notes, vol. 10. Aarhus University (2001).
Bovier, A.: Statistical mechanics of disordered systems, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 18. Cambridge University Press, Cambridge (2006). A mathematical perspective.
Bovier, A., Faggionato, A.: Spectral characterization of aging: the REM-like trap model. Ann. Appl. Probab. 15(3), 1997–2037 (2005).
Bovier, A., Kurkova, I.: Local energy statistics in disordered systems: a proof of the local REM conjecture. Comm. Math. Phys. 263(2), 513–533 (2006).
Bovier, A., Kurkova, I.: Local energy statistics in spin glasses. J. Stat. Phys. 126(4–5), 933–949 (2007).
Černý, J., Gayrard, V.: Hitting time of large subsets of the hypercube. Random Struct. Algorithms 33(2), 252–267 (2008).
Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24(5), 2613–2626 (1981).
Georgii, H.O.: Large deviations and maximum entropy principle for interacting random fields on Z d. Ann. Probab. 21(4), 1845–1875 (1993).
Kallenberg, O.: Random measures, third edn. Akademie-Verlag, Berlin (1983).
Kuptsov, A.: Universality of random Hamiltonians. Ph.D. thesis, New York University (2008).
Kurkova, I.: Local energy statistics in directed polymers. Electron. J. Probab. 13, no. 2, 5–25 (2008).
Mertens, S.: Random costs in combinatorial optimization. Phys. Rev. Lett. 84(6), 1347–1350 (2000).
Talagrand, M.: Spin glasses: a challenge for mathematicians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 46. Springer-Verlag, Berlin (2003). Cavity and mean field models.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Arous, G.B., Kuptsov, A. (2009). REM Universality for Random Hamiltonians. In: de Monvel, A.B., Bovier, A. (eds) Spin Glasses: Statics and Dynamics. Progress in Probability, vol 62. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9891-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-7643-9891-0_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8999-4
Online ISBN: 978-3-7643-9891-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)