Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models

Bernard Derrida; Peter Mottishaw

doi:10.5802/crphys.199

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Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models
[Généralisations de la brisure de symétrie des répliques de Parisi et des overlaps dans les modèles d’énergies aléatoires]

Bernard Derrida ^1,² ; Peter Mottishaw ³

¹ Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France
² Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
³ SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

Comptes Rendus. Physique, Volume 25 (2024), pp. 329-351.

Cet article fait partie du numéro thématique Gérard Toulouse, une vie de découvertes et d'engagement coordonné par Bernard Derrida et al..

Résumé
Abstract

Le modèle d’énergies aléatoires (REM) est le modèle de verre de spin le plus simple qui présente une brisure de symétrie des répliques. Il est bien connu depuis les années 80 que ses overlaps ne sont pas automoyennants et que leurs statistiques sont celles prédites par la méthode des répliques. Ces propriétés statistiques peuvent être comprises en considérant que les niveaux d’énergie les plus bas sont les points générés par un processus de Poisson de densité exponentielle. Nous montrons ici dans un premier temps comment ces statistiques d’overlaps sont modifiées lorsqu’on remplace la densité exponentielle par une somme de deux exponentielles. Une façon de concilier ces résultats avec la théorie des répliques est de permettre aux blocs de la matrice de Parisi de fluctuer. D’autres exemples où la taille de ces blocs doit fluctuer incluent les corrections de taille finie du REM, le cas des énergies discrètes et les overlaps entre deux températures. Dans tous ces cas, non seulement la taille des blocs fluctue mais elle doit prendre des valeurs complexes si l’on souhaite reproduire nos résultats obtenus directement, c’est à dire sans utiliser la méthode des répliques.

The random energy model (REM) is the simplest spin glass model which exhibits replica symmetry breaking. It is well known since the 80’s that its overlaps are non-selfaveraging and that their statistics satisfy the predictions of the replica theory. All these statistical properties can be understood by considering that the low energy levels are the points generated by a Poisson process with an exponential density. Here we first show how, by replacing the exponential density by a sum of two exponentials, the overlaps statistics are modified. One way to reconcile these results with the replica theory is to allow the blocks in the Parisi matrix to fluctuate. Other examples where the sizes of these blocks should fluctuate include the finite size corrections of the REM, the case of discrete energies and the overlaps between two temperatures. In all these cases, the block sizes not only fluctuate but need to take complex values if one wishes to reproduce the results of our replica-free calculations.

Reçu le : 2024-06-30
Accepté le : 2024-08-07
Publié le : 2024-09-24

DOI : 10.5802/crphys.199

Keywords: Disordered systems, Spin glasses, Replica symmetry breaking, Random Energy Model
Mot clés : Systèmes désordonnés, Verres de spin, Brisure de symétrie des répliques, Modèle d’énergies aléatoires

Affiliations des auteurs :

Bernard Derrida ^{1, 2} ; Peter Mottishaw ³

¹ Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France
² Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
³ SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Bernard Derrida; Peter Mottishaw. Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models. Comptes Rendus. Physique, Volume 25 (2024), pp. 329-351. doi : 10.5802/crphys.199. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.199/

Bibliographie
Cité par

[1] S. F. Edwards; P. W. Anderson Theory of spin glasses, J. Phys. F: Met. Phys., Volume 5 (1975) no. 5, p. 965 | DOI

[2] G. Parisi The Overlap in Glassy Systems, Stealing the Gold: A celebration of the pioneering physics of Sam Edwards (D. Sherrington; P. Goldbart; N. Goldenfeld, eds.) (International Series of Monographs on Physics), Volume 126, Oxford University Press, 2004, pp. 192-211 | DOI

[3] D. Sherrington; S. Kirkpatrick Solvable model of a spin-glass, Phys. Rev. Lett., Volume 35 (1975) no. 26, pp. 1792-1796 | DOI

[4] G. Toulouse; B. Derrida Free energy probability distribution in the SK spin glass model, Proceedings of the Sixth Brazillian Symposium on Theoretical Physics (E. M. Ferreira; B. Koiller, eds.), Conselho Nacional de Desenvolvimento Científico e Tecnológico (1981), pp. 143-171

[5] G. Parisi Order parameter for spin-glasses, Phys. Rev. Lett., Volume 50 (1983) no. 24, pp. 1946-1948 | DOI

[6] M. Mézard; G. Parisi; N. Sourlas; G. Toulouse; M. A. Virasoro Nature of the spin-glass phase, Phys. Rev. Lett., Volume 52 (1984) no. 13, pp. 1156-1159 | DOI

[7] M. Mézard; G. Parisi; N. Sourlas; G. Toulouse; M. A. Virasoro Replica symmetry breaking and the nature of the spin glass phase, J. Phys., Volume 45 (1984) no. 5, pp. 843-854 | DOI | Zbl

[8] M. Mézard; G. Parisi; M. A. Virasoro Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications, World Scientific Lecture Notes In Physics, 9, World Scientific, 1987 | DOI

[9] F. Guerra Broken replica symmetry bounds in the mean field spin glass model, Commun. Math. Phys., Volume 233 (2003) no. 1, pp. 1-12 | DOI

[10] M. Talagrand The Parisi formula, Ann. Math., Volume 163 (2006) no. 1, pp. 221-263 | DOI | Zbl

[11] L.-P. Arguin; M. Aizenman On the structure of quasi-stationary competing particle systems, Ann. Probab., Volume 37 (2009) no. 3, pp. 1080-1113 | DOI | Zbl

[12] D. Panchenko The Parisi ultrametricity conjecture, Ann. Math., Volume 177 (2013) no. 1, pp. 383-393 | DOI | Zbl

[13] Spin glass theory and far beyond. Replica Symmetry Breaking After 40 Years (P. Charbonneau; E. Marinari; M. Mézard; G. Parisi; F. Ricci-Tersenghi; G. Sicuro; F. Zamponi, eds.), World Scientific, 2023 | DOI

[14] B. Derrida; H. Flyvbjerg Statistical properties of randomly broken objects and of multivalley structures in disordered systems, J. Phys. A. Math. Gen., Volume 20 (1987) no. 15, pp. 5273-5288 | DOI

[15] S. Ghirlanda; F. Guerra General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A. Math. Gen., Volume 31 (1998) no. 46, pp. 9149-9155 | DOI

[16] B. Derrida; G. Toulouse Sample to sample fluctuations in the random energy model, J. Physique Lett., Volume 46 (1985) no. 6, pp. 223-228 | DOI

[17] B. Derrida Random-energy model: Limit of a family of disordered models, Phys. Rev. Lett., Volume 45 (1980) no. 2, pp. 79-82 | DOI

[18] B. Derrida Random-energy model: An exactly solvable model of disordered systems, Phys. Rev. B, Volume 24 (1981) no. 5, pp. 2613-2626 | DOI

[19] T. Eisele On a third-order phase transition, Commun. Math. Phys., Volume 90 (1983) no. 1, pp. 125-159 | DOI | Zbl

[20] A. Galves; S. Martinez; P. Picco Fluctuations in Derrida’s random energy and generalized random energy models, J. Stat. Phys., Volume 54 (1989) no. 1, pp. 515-529 | DOI

[21] E. Olivieri; P. Picco On the existence of thermodynamics for the random energy model, Commun. Math. Phys., Volume 96 (1984) no. 1, pp. 125-144 | DOI | Zbl

[22] A. Bovier; I. Kurkova; M. Löwe Fluctuations of the free energy in the REM and the p-spin SK models, Ann. Probab., Volume 30 (2002) no. 2, pp. 605-651 | DOI | Zbl

[23] E. Bolthausen Random media and spin glasses: An introduction into some mathematical results and problems, Spin glasses (E. Bolthausen; A. Bovier, eds.) (Lecture Notes in Mathematics), Volume 1900, Springer, 2007, pp. 1-44 | DOI | Zbl

[24] N. Kistler Derrida’s random energy models. From Spin Glasses to the Extremes of Correlated Random Fields, Correlated random systems: five different methods (Lecture Notes in Mathematics), Volume 2143, Springer; Société Mathématique de France, Paris, 2015, pp. 71-120 | DOI | Zbl

[25] L. A. Pastur A limit theorem for sums of exponentials, Math. Notes, Volume 46 (1989) no. 3, pp. 712-716 | DOI | Zbl

[26] G. Ben Arous; L. V. Bogachev; S. A. Molchanov Limit theorems for sums of random exponentials, Probab. Theory Rel., Volume 132 (2005) no. 4, pp. 579-612 | DOI | Zbl

[27] B. Derrida; P. Mottishaw; V. Gayrard Random energy models: Broken replica symmetry and activated dynamics, World Scientific (2023), pp. 657-677 | DOI

[28] M. Campellone Some non-perturbative calculations on spin glasses, J. Phys. A. Math. Gen., Volume 28 (1995) no. 8, pp. 2149-2158 | DOI | Zbl

[29] M. Campellone; G. Parisi; M. A. Virasoro Replica method and finite volume corrections, J. Stat. Phys., Volume 138 (2009) no. 1-3, pp. 29-39 | DOI | Zbl

[30] B. Derrida; P. Mottishaw Finite size corrections in the random energy model and the replica approach, J. Stat. Mech. Theory Exp., Volume 2015 (2015) no. 1, P01021 | DOI | Zbl

[31] C. Moukarzel; N. Parga Numerical complex zeros of the random energy model, Phys. A: Stat. Mech. Appl., Volume 177 (1991) no. 1-3, pp. 24-30 | DOI

[32] B. Derrida The zeroes of the partition function of the random energy model, Phys. A: Stat. Mech. Appl., Volume 177 (1991) no. 1-3, pp. 31-37 | DOI

[33] D. B. Saakian Random energy model at complex temperatures, Phys. Rev. E, Volume 61 (2000) no. 6, pp. 6132-6135 | DOI

[34] G. Bunin; L. Foini; J. Kurchan Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems (2023) (preprint, arXiv:2207.02473) | DOI

[35] K. Ogure; Y. Kabashima An exact analytic continuation to complex replica number in the discrete random energy model of finite system size, Prog. Theor. Phys. Supp., Volume 157 (2005), pp. 103-106 | DOI

[36] N. K. Jana Contributions to random energy models, Ph. D. Thesis, Indian Statistical Institute-Kolkata, India (2007) | arXiv | DOI

[37] B. Derrida; P. Mottishaw The discrete random energy model and one step replica symmetry breaking, J. Phys. A. Math. Theor., Volume 55 (2022) no. 26, 265002 | DOI | Zbl

[38] E. Gardner; B. Derrida The probability distribution of the partition function of the random energy model, J. Phys. A. Math. Gen., Volume 22 (1989) no. 12, pp. 1975-1981 | DOI

[39] B. Derrida From random walks to spin glasses, Phys. D: Nonlinear Phenom., Volume 107 (1997) no. 2-4, pp. 186-198 | DOI | Zbl

[40] D. J. Gross; M. Mézard The simplest spin glass, Nucl. Phys., B, Volume 240 (1984) no. 4, pp. 431-452 | DOI

[41] E. Gardner Spin glasses with p-spin interactions, Nucl. Phys., B, Volume 257 (1985), pp. 747-765 | DOI

[42] G. Parisi Infinite number of order parameters for spin-glasses, Phys. Rev. Lett., Volume 43 (1979), pp. 1754-1756 | DOI

[43] J.-P. Bouchaud; F. Krzakala; O. C. Martin Energy exponents and corrections to scaling in Ising spin glasses, Phys. Rev. B, Volume 68 (2003) no. 22, 224404 | DOI

[44] A. J. Bray; M. A. Moore Chaotic nature of the spin-glass phase, Phys. Rev. Lett., Volume 58 (1987) no. 1, pp. 57-60 | DOI

[45] T. Rizzo Chaos in mean-field spin-glass models, Spin Glasses: Statics and Dynamics (A. Boutet de Monvel; Anton Bovier, eds.) (Progress in Probability), Volume 62, Birkhäuser (2009), pp. 143-157 | DOI | Zbl

[46] M. Sales; J.-P. Bouchaud Rejuvenation in the random energy model, Eur. Phys. Lett., Volume 56 (2001) no. 2, p. 181 | DOI

[47] F. Krzakala; O. C. Martin Chaotic temperature dependence in a model of spin glasses, Eur. Phys. J. B, Condens. Matter Complex Syst., Volume 28 (2002) no. 2, pp. 199-208 | DOI

[48] I. Kurkova Temperature Dependence of the Gibbs State in the Random Energy Model, J. Stat. Phys., Volume 111 (2003) no. 1, pp. 35-56 | DOI | Zbl

[49] M. Pain; O. Zindy Two-temperatures overlap distribution for the 2D discrete Gaussian free field, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 57 (2021) no. 2, pp. 685-699 | DOI | Zbl

[50] B. Derrida; P. Mottishaw One step replica symmetry breaking and overlaps between two temperatures, J. Phys. A. Math. Theor., Volume 54 (2021) no. 4, 045002 | DOI | Zbl

[51] R. B. Paris; D. Kaminski Asymptotics and Mellin–Barnes integrals, Encyclopedia of Mathematics and Its Applications, 85, Cambridge University Press, 2001 | DOI

[52] A. Crisanti; H.-J. Sommers The spherical $p$ -spin interaction spin glass model: the statics, Z. Phys., B, Volume 87 (1992) no. 3, pp. 341-354 | DOI

[53] B. Derrida; H. Spohn Polymers on disordered trees, spin glasses, and traveling waves, J. Stat. Phys., Volume 51 (1988) no. 5, pp. 817-840 | DOI | Zbl

[54] T. Obuchi; Y. Kabashima; H. Nishimori Complex replica zeros of $\pm$ J Ising spin glass at zero temperature, J. Phys. A. Math. Theor., Volume 42 (2009) no. 7, 075004 | DOI

[55] E. Gardner; B. Derrida Optimal storage properties of neural network models, J. Phys. A. Math. Gen., Volume 21 (1988) no. 1, pp. 271-284 | DOI

[56] E. Gardner; B. Derrida Three unfinished works on the optimal storage capacity of networks, J. Phys. A. Math. Gen., Volume 22 (1989) no. 12, pp. 1983-1994 | DOI

[57] W. Krauth; M. Mézard Storage capacity of memory networks with binary couplings, J. Phys. (Paris), Volume 50 (1989) no. 20, pp. 3057-3066 | DOI

[58] H. Huang; Y. Kabashima Origin of the computational hardness for learning with binary synapses, Phys. Rev. E, Volume 90 (2014) no. 5, 052813 | DOI

[59] J. Ding; N. Sun Capacity Lower Bound for the Ising Perceptron, Proceedings of the 51-ST. Annual ACM SIGACTSymposium on Theory of Computing (STOC ‘19) (M. Charikar; E. Cohen, eds.), ACM Press (2019), pp. 816-827 | DOI

[60] R. Monasson; R. Zecchina Entropy of the K-Satisfiability Problem, Phys. Rev. Lett., Volume 76 (1996) no. 21, pp. 3881-3885 | DOI | Zbl

[61] R. Monasson; R. Zecchina Statistical mechanics of the random K-satisfiability model, Phys. Rev. E, Volume 56 (1997) no. 2, pp. 1357-1370 | DOI

[62] M. Mezard; A. Montanari Information, physics, and computation, Oxford Graduate Texts, Oxford University Press, 2009, pp. 3-22 | DOI | Zbl

[63] B. Derrida A generalization of the random energy model which includes correlations between energies, J. Physique Lett., Volume 46 (1985) no. 9, pp. 401-407 | DOI

[64] D. Ruelle A mathematical reformulation of Derrida’s REM and GREM, Commun. Math. Phys., Volume 108 (1987) no. 2, pp. 225-239 | DOI | Zbl

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