Much ado about Derrida's GREM

Lecture Notes in Mathematics

Volumn 1900, Issue , 2007, Pages 81-115

  Bovier, Anton ^a,b   Kurkova, Irina ^c  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^c SORBONNE UNIVERSITÉ   (France)

Author keywords

Branching processes; Coalescence; Gaussian processes; Generalised random energy model; Poisson point processes; Spin glasses

Indexed keywords

EID: 33846208628     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-540-40908-3_4     Document Type: Article

References (36)

1. F. Guerra, Broken replica symmetry bounds in the mean field spin-glass model, Commun. Math. Phys. 233, 1-12 (2003)
2. M. Aizenman, R. Sims, and S.L. Starr. An extended variational principle for the SK spin-glass model. Phys. Rev. B, 6821, 4403 (2003)
3. M. Talagrarid, The generalized Parisi formula. C. R. Math. Acad. Sci. Paris 337, 111-114 (2003)
4. M. Talagraud, The Parisi formula, to appear in Ann. Math. 163, 221-263 (2006)
5. M. Talagrand, Self organization in the low-temperature region of a spin-glass model, Rev. Math. Phys. 15, 1-78 (2003)
6. M. Talagrand, Spin-Glasses: A Challenge to Mathematicians, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 46, Springer, Berlin Heidelberg New York (2003)
7. B. Derrida. A generalisation of the random energy model that includes correlations between the energies. J. Phys. Lett. 46, 401-407 (1985)
8. B. Derrida and E. Gardner. Solution of the generalised random energy model. J. Phys. C 19, 2253-2274 (1986)
9. B. Derrida and E. Gardner. Magnetic properties and function q(x) of the generalised random, energy model. J. Phys. C 19, 5783-5798 (1986)
10. A. Bovier and I. Kurkova. Derrida's generalised random energy models 1: Poisson cascades and Gibbs measures. Ann. de l'I.H.P. 40, 439-480 (2004)
11. A. Bovier and I. Kurkova. Derrida's generalised random energy models 2: Models with continuous hierarchies. Ann. de l'I.H.P. 40, 481-495 (2004)
12. A. Bovier and I. Kurkova. Gibbs measures of Derrida's generalised random, energy models and the genealogy of Neveu's continuous state branching process. Preprint, University Paris 6 (2003)
13. D. Sherrington and S. Kirkpatrick. Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792-1796 (1972)
14. B. Derrida. Random energy model: Limit, of a family of disordered models, Phys. Rev. Letts. 45, 79-82 (1980)
15. B. Derrida. Random energy model: An exactly solvable model of disordered systems, Phys. Rev. B 24, 2613-2626 (1981)
16. E. Gardner and B. Derrida. The probability distribution of the partition function of the random energy model. J. Phys. A 22, 1975-1981 (1989)
17. D. Capocaccia, M. Cassandro, and P. Picco. On the existence of thermodynamics for the generalised random energy models. J. Stat. Phys. 46, 493-505 (1987)
18. T.H. Eisele, On a third order phase transition, Commun. Math. Phys. 90, 125-159 (1983)
19. A. Galvez, S. Martinez, and P. Picco. Fluctuations in Derrida's random, energy and generalised random energy models. J. Stat. Phys. 54, 515-529 (1989)
20. D. Ruelle. A mathematical reformulation of Derrida's REM and GREM. Commun. Math. Phys. 108, 225-239 (1987)
21. J. Neveu. A continuous state branching process in relation with the GREM model of spin-glass theory. Rapport interne no. 267, Ecole Polytechnique
22. E. Bolthausen and A.S. Sznitman. On Ruelle's probability cascades and an abstract, cavity method. Commun. Math. Phys. 107, 247-276 (1998)
23. J. Bertoin and J.F. Le Gall. The Bolthausen-Sznitman coalescent and the genealogy of continuous state branching processes. Prob. Theory Relat. Fields 117, 249-266 (2000)
24. A. Bovier, I.A.Kurkova, and M. Löwe. Fluctuations of the free energy in the REM and p-spin SK models. Ann. Probab. 30(2), 605-651 (2002)
25. A. Bovier. Statistical mechanics of disordered systems, MaPhySto Lecture Notes 10, Aarhus, (2001). In expanded and updated form: Cambridge Series in Statistical and Probabilistic Mathematics, vol. 18, Cambridge University Press, Cambridge 2006
26. E. Bolthausen and A.-S. Sznitman, Ten Lectures on Random Media, Birkhäuser, Basel (2002)
27. M. Talagrand, Mean field models for spin-glasses: a first course. Lectures on probability theory and statistics (Saint-Flour, 2000), 181-285, Lecture Notes in Mathematics, 1816, Springer, Berlin Heidelberg New York, 2003
28. S. Ghirlanda and F. Guerra. General properties of the overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A 31, 9144-9155 (1998)
29. M. Talagrand, Rigorous low temperature results for mean field p-spin interaction models, Prob. Theory. Relat. Fields 117, 303-360 (2000)
30. M.R. Leadbetter, G. Lindgren, and H. Rootzén. Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin Heidelberg New York (1983)
31. A. Bovier and I. Kurkova. Rigorous results on some simple spin-glass models. Markov Process. Relat. Fields 9 (2), 209-242 (2003)
32. M. Ledoux and M. Talagrand, Probability on Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, Springer, Berlin Heidelberg New York (1991)
33. M.D. Bramson. Maximal displacement of branching Brownian motion, Commun. Pure Appl. Math. 31, 531-581 (1978)
34. J. Pitman. Combinatorial stochastic processes. Lecture Notes of the 2002 Ecole do Probabilités de St. Flour, Springer Lecture Notes in Mathematics
35. J. Pitman, Exchangeable and partially exchangeable random partitions, Probab. Theory. Relat. Fields 102, 145-158 (1995)
36. M. Perrnan, J. Pitman, and M. Yor, Size-biased sampling of Poisson point processes and excursions, Probab. Theory. Relat. Fields 92, 21-39 (1992)