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.
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.
Accepted:
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Mot clés : Systèmes désordonnés, Verres de spin, Brisure de symétrie des répliques, Modèle d’énergies aléatoires
Bernard Derrida 1, 2; Peter Mottishaw 3
@article{CRPHYS_2024__25_G1_329_0, author = {Bernard Derrida and Peter Mottishaw}, title = {Generalizations of {Parisi{\textquoteright}s} replica symmetry breaking and overlaps in random energy models}, journal = {Comptes Rendus. Physique}, pages = {329--351}, publisher = {Acad\'emie des sciences, Paris}, volume = {25}, year = {2024}, doi = {10.5802/crphys.199}, language = {en}, }
TY - JOUR AU - Bernard Derrida AU - Peter Mottishaw TI - Generalizations of Parisi’s replica symmetry breaking and overlaps in random energy models JO - Comptes Rendus. Physique PY - 2024 SP - 329 EP - 351 VL - 25 PB - Académie des sciences, Paris DO - 10.5802/crphys.199 LA - en ID - CRPHYS_2024__25_G1_329_0 ER -
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/
[1] Theory of spin glasses, J. Phys. F: Met. Phys., Volume 5 (1975) no. 5, p. 965 | DOI
[2] 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] Solvable model of a spin-glass, Phys. Rev. Lett., Volume 35 (1975) no. 26, pp. 1792-1796 | DOI
[4] 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] Order parameter for spin-glasses, Phys. Rev. Lett., Volume 50 (1983) no. 24, pp. 1946-1948 | DOI
[6] Nature of the spin-glass phase, Phys. Rev. Lett., Volume 52 (1984) no. 13, pp. 1156-1159 | DOI
[7] Replica symmetry breaking and the nature of the spin glass phase, J. Phys., Volume 45 (1984) no. 5, pp. 843-854 | DOI | Zbl
[8] 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] Broken replica symmetry bounds in the mean field spin glass model, Commun. Math. Phys., Volume 233 (2003) no. 1, pp. 1-12 | DOI
[10] The Parisi formula, Ann. Math., Volume 163 (2006) no. 1, pp. 221-263 | DOI | Zbl
[11] On the structure of quasi-stationary competing particle systems, Ann. Probab., Volume 37 (2009) no. 3, pp. 1080-1113 | DOI | Zbl
[12] 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] 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] 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] Sample to sample fluctuations in the random energy model, J. Physique Lett., Volume 46 (1985) no. 6, pp. 223-228 | DOI
[17] Random-energy model: Limit of a family of disordered models, Phys. Rev. Lett., Volume 45 (1980) no. 2, pp. 79-82 | DOI
[18] Random-energy model: An exactly solvable model of disordered systems, Phys. Rev. B, Volume 24 (1981) no. 5, pp. 2613-2626 | DOI
[19] On a third-order phase transition, Commun. Math. Phys., Volume 90 (1983) no. 1, pp. 125-159 | DOI | Zbl
[20] Fluctuations in Derrida’s random energy and generalized random energy models, J. Stat. Phys., Volume 54 (1989) no. 1, pp. 515-529 | DOI
[21] On the existence of thermodynamics for the random energy model, Commun. Math. Phys., Volume 96 (1984) no. 1, pp. 125-144 | DOI | Zbl
[22] 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] 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] 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] A limit theorem for sums of exponentials, Math. Notes, Volume 46 (1989) no. 3, pp. 712-716 | DOI | Zbl
[26] Limit theorems for sums of random exponentials, Probab. Theory Rel., Volume 132 (2005) no. 4, pp. 579-612 | DOI | Zbl
[27] Random energy models: Broken replica symmetry and activated dynamics, World Scientific (2023), pp. 657-677 | DOI
[28] Some non-perturbative calculations on spin glasses, J. Phys. A. Math. Gen., Volume 28 (1995) no. 8, pp. 2149-2158 | DOI | Zbl
[29] Replica method and finite volume corrections, J. Stat. Phys., Volume 138 (2009) no. 1-3, pp. 29-39 | DOI | Zbl
[30] 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] Numerical complex zeros of the random energy model, Phys. A: Stat. Mech. Appl., Volume 177 (1991) no. 1-3, pp. 24-30 | DOI
[32] 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] Random energy model at complex temperatures, Phys. Rev. E, Volume 61 (2000) no. 6, pp. 6132-6135 | DOI
[34] Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems (2023) (preprint, arXiv:2207.02473) | DOI
[35] 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] Contributions to random energy models, Ph. D. Thesis, Indian Statistical Institute-Kolkata, India (2007) | arXiv | DOI
[37] 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] 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] From random walks to spin glasses, Phys. D: Nonlinear Phenom., Volume 107 (1997) no. 2-4, pp. 186-198 | DOI | Zbl
[40] The simplest spin glass, Nucl. Phys., B, Volume 240 (1984) no. 4, pp. 431-452 | DOI
[41] Spin glasses with p-spin interactions, Nucl. Phys., B, Volume 257 (1985), pp. 747-765 | DOI
[42] Infinite number of order parameters for spin-glasses, Phys. Rev. Lett., Volume 43 (1979), pp. 1754-1756 | DOI
[43] Energy exponents and corrections to scaling in Ising spin glasses, Phys. Rev. B, Volume 68 (2003) no. 22, 224404 | DOI
[44] Chaotic nature of the spin-glass phase, Phys. Rev. Lett., Volume 58 (1987) no. 1, pp. 57-60 | DOI
[45] 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] Rejuvenation in the random energy model, Eur. Phys. Lett., Volume 56 (2001) no. 2, p. 181 | DOI
[47] 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] 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] 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] One step replica symmetry breaking and overlaps between two temperatures, J. Phys. A. Math. Theor., Volume 54 (2021) no. 4, 045002 | DOI | Zbl
[51] Asymptotics and Mellin–Barnes integrals, Encyclopedia of Mathematics and Its Applications, 85, Cambridge University Press, 2001 | DOI
[52] The spherical -spin interaction spin glass model: the statics, Z. Phys., B, Volume 87 (1992) no. 3, pp. 341-354 | DOI
[53] Polymers on disordered trees, spin glasses, and traveling waves, J. Stat. Phys., Volume 51 (1988) no. 5, pp. 817-840 | DOI | Zbl
[54] Complex replica zeros of J Ising spin glass at zero temperature, J. Phys. A. Math. Theor., Volume 42 (2009) no. 7, 075004 | DOI
[55] Optimal storage properties of neural network models, J. Phys. A. Math. Gen., Volume 21 (1988) no. 1, pp. 271-284 | DOI
[56] 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] Storage capacity of memory networks with binary couplings, J. Phys. (Paris), Volume 50 (1989) no. 20, pp. 3057-3066 | DOI
[58] Origin of the computational hardness for learning with binary synapses, Phys. Rev. E, Volume 90 (2014) no. 5, 052813 | DOI
[59] 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] Entropy of the K-Satisfiability Problem, Phys. Rev. Lett., Volume 76 (1996) no. 21, pp. 3881-3885 | DOI | Zbl
[61] Statistical mechanics of the random K-satisfiability model, Phys. Rev. E, Volume 56 (1997) no. 2, pp. 1357-1370 | DOI
[62] Information, physics, and computation, Oxford Graduate Texts, Oxford University Press, 2009, pp. 3-22 | DOI | Zbl
[63] 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] 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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