Derrida's generalised random energy models 1 : models with finitely many hierarchies

Bovier, Anton; Kurkova, Irina

doi:10.1016/j.anihpb.2003.09.002

Bovier, Anton ; Kurkova, Irina

Annales de l'I.H.P. Probabilités et statistiques, Tome 40 (2004) no. 4, pp. 439-480.

MR Zbl | 2 citations dans Numdam

DOI : 10.1016/j.anihpb.2003.09.002

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     title = {Derrida's generalised random energy models 1 : models with finitely many hierarchies},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {439--480},
     publisher = {Elsevier},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpb.2003.09.002/}
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Bovier, Anton; Kurkova, Irina. Derrida's generalised random energy models 1 : models with finitely many hierarchies. Annales de l'I.H.P. Probabilités et statistiques, Tome 40 (2004) no. 4, pp. 439-480. doi : 10.1016/j.anihpb.2003.09.002. http://archive.numdam.org/articles/10.1016/j.anihpb.2003.09.002/

Bibliographie
Cité par

[1] M Aizenmann, P Contucci, On the stability of the quenched state in mean field spin-glass models, J. Statist. Phys. 92 (1998) 765-783. | MR | Zbl

[2] A Bovier, Statistical Mechanics of Disordered Systems, MaPhySto Lecture Notes, vol. 10, Aarhus, 2001.

[3] A Bovier, V Gayrard, The Hopfield model as a generalised random mean field model, in: Bovier A, Picco P (Eds.), Mathematics of Spin Glasses and Neural Networks, Progress in Probablity, Birkhäuser, Boston, 1997. | Zbl

[4] A. Bovier, I. Kurkova, Derrida's generalised random energy models 2. Models with continuous hierarchies, Ann. IHP (2002), submitted for publication. | Numdam | Zbl

[5] A. Bovier, I. Kurkova, Gibbs measures of Derrida's generalised random energy models and geneologies of Neveu's continuous state branching process, Probab. Theor. Relat. Fields (2003), submitted for publication.

[6] A Bovier, I Kurkova, M Löwe, Fluctuations of the free energy in the REM and the p-spin SK-models, Ann. Probab. 30 (2002) 605-651. | MR | Zbl

[7] M.D Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978) 531-581. | MR | Zbl

[8] D Capocaccia, M Cassandro, P Picco, On the existence of thermodynamics for the generalised random energy model, J. Statist. Phys. 46 (1987) 493-505. | MR

[9] B Derrida, Random energy model: limit of a family of disordered models, Phys. Rev. Lett. 45 (1980) 79-82. | MR

[10] B Derrida, Random energy model: an exactly solvable model of disordered systems, Phys. Rev. B 24 (1981) 2613-2626. | MR

[11] B Derrida, A generalisation of the random energy model that includes correlations between the energies, J. Phys. Lett. 46 (1985) 401-407.

[12] B Derrida, E Gardner, Solution of the generalised random energy model, J. Phys. C 19 (1986) 2253-2274.

[13] B Derrida, E Gardner, Magnetic properties and function q(x) of the generalised random energy model, J. Phys. C 19 (1986) 5783-5798.

[14] E Gardner, B Derrida, The probability distribution of the partition function of the random energy model, J. Phys. A 22 (1989) 1975-1981. | MR

[15] D.J Daley, D Vere-Jones, An Introduction to the Theory of Point Processes, Springer, New York, 1988. | MR | Zbl

[16] T.C Dorlas, J.R Wedagedera, Large deviations and the random energy model, Int. J. Mod. Phys. B 15 (2001) 1-15. | MR

[17] T.H Eisele, On a third order phase transition, Comm. Math. Phys. 90 (1983) 125-159. | MR | Zbl

[18] W Feller, An Introduction to Probability Theory and its Applications, vol. I, Wiley, New York, 1966. | MR | Zbl

[19] S Ghirlanda, F Guerra, General properties of the overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A 31 (1998) 9144-9155. | MR | Zbl

[20] A Galvez, S Martinez, P Picco, Fluctuations in Derrida's random energy and generalised random energy models, J. Statist. Phys. 54 (1989) 515-529. | MR

[21] O Kallenberg, Random Measures, Akademie Verlag, Berlin, 1986. | MR | Zbl

[22] M.F. Kratz, P. Picco, On a representation of Gibbs measure for R.E.M., preprint Samos 151, 2002.

[23] I Kurkova, Temperature dependence of the Gibbs State in the random energy model, J. Stat. Phys. 111 (1/2) (2003) 35-56. | MR | Zbl

[24] M.R Leadbetter, G Lindgren, H Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer, Berlin, 1983. | MR | Zbl

[25] M Ledoux, L Talagrand, Probability in Banach Space, Springer, Berlin, 1991. | MR | Zbl

[26] J. Neveu, A continuous state branching process in relation with the GREM model of spin glass theory, Rapport interne no 267, École Polytechnique, 1992.

[27] E Olivieri, P Picco, On the existence of thermodynamics for the random energy model, Comm. Math. Phys. 96 (1991) 125-144. | MR | Zbl

[28] S.I Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987. | MR | Zbl

[29] R.T Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. | MR | Zbl

[30] D Ruelle, A mathematical reformulation of Derrida's REM and GREM, Comm. Math. Phys. 108 (1987) 225-239. | MR | Zbl

[31] D Sherrington, S Kirkpatrick, Solvable model of a spin glass, Phys. Rev. Lett. 35 (1972) 1792-1796.

[32] M Talagrand, Concentration of measure and isoperimetric inequalities in product space, Publ. Math. IHES 81 (1995) 73-205. | Numdam | MR | Zbl

[33] M Talagrand, Rigorous results for the Hopfield model with many patterns, Probab. Theory Related Fields 110 (1998) 177-276. | MR | Zbl

[34] M. Talagrand, Mean Field Models for Spin Glasses: a First Course, Lecture Notes of the 2000 École des Probabilités de St. Flour, Lecture Notes in Mathematics, Springer, submitted for publication. | MR | Zbl

[35] M Talagrand, Self organisation in the low-temperature region of a spin glass model, Rev. Math. Phys. 15 (2003) 1-78. | MR | Zbl

[36] M Talagrand, Spin Glasses: a Challenge to Mathematicians, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 46, Springer, Berlin, 2003. | MR | Zbl

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