@article{AIHPA_1996__64_3_255_0,
author = {Petritis, Dimitri},
title = {Equilibrium statistical mechanics of frustrated spin glasses : a survey of mathematical results},
journal = {Annales de l'I.H.P. Physique th\'eorique},
volume = {65},
year = {1996},
pages = {255-288},
mrnumber = {1400292},
zbl = {0853.60081},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPA_1996__64_3_255_0}
}
Petritis, Dimitri. Equilibrium statistical mechanics of frustrated spin glasses : a survey of mathematical results. Annales de l'I.H.P. Physique théorique, Tome 65 (1996) pp. 255-288. http://gdmltest.u-ga.fr/item/AIHPA_1996__64_3_255_0/
[1] , and , Some rigorous results on the Sherrington-Kirkpatrick model, Commun. Math. Phys., Vol. 112, 1987, pp. 3-20. | MR 904135
[2] and , Rounding of first order phase transitions in systems with quenched disorder, Commun. Math. Phys., Vol. 130, 1990, pp. 489-528. | MR 1060388 | Zbl 0714.60090
[3] and , Stability of the Sherrington-Kirkpatrick solution of a spin glass model, J. Phys. A. Math. Gen., Vol. 11, 1978, pp. 983-990.
[4] , and , Random infinite volume Gibbs states for the Curie-Weiss random field Ising model, J. Stat. Phys., Vol. 66, 1992, pp. 139-164. | MR 1149483 | Zbl 0925.60127
[5] , and , Spin-glass models of neural networks, Phys. Rev., A32, 1985, pp. 1007-1018. | MR 797031
[6] and , Uniqueness of a Gibbs field with random potential: an elementary approach, Th. Prob. Appl., Vol. 31, 1988, pp. 572-589. | Zbl 0635.60107
[7] and , Huge symmetric Langevin spin glass dynamics, preprint Université de Paris XI, 1994. | MR 1457623
[8] , and , Strict inequalities for critical values of Potts models and random chluster processes, preprint Bristol Université;, 1992.
[9] and , Spin glasses: experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys., Vol. 58, 1988, pp. 801-976.
[10] , and , Interface roughening and random field instabilities, Phys. Rev., B24, 1981, pp. 6736-6739.
[11] , The Bethe lattice spin glass at zero temperature, Ann. Inst. H. Poincaré, Vol. 54, 1991, pp. 89-113. | Numdam | MR 1102973 | Zbl 0719.60123
[12] , Disordered systems and random geometry: polymers, spin glasses, interfaces, Ph. D. dissertation, ETH Zurich, 1986.
[13] and , A heuristic theory of the spin glass phase, J. Stat. Phys., Vol. 44, 1986, pp. 347-391. | MR 857062
[14] , and , Gibbs states of the Hopfield model with extensively many patterns, preprint Weiersrass IAAS, Berlin, 1994. | MR 1325589
[15] , and , Large deviation principales for the Hopfield and Kac-Hopfield models, preprint CPT Marseille, 1994. | MR 1305586
[16] and , Phase transition in the three-dimensional random field Ising model, Commun. Math. Phys., Vol. 116, 1988, pp. 539-572. | MR 943702
[17] and , Random walk in asymmetric random environments, Commun. Math. Phys., Vol. 142, 1991, pp. 345-420. | MR 1137068 | Zbl 0734.60112
[18] , and , Directed polymers on trees: a martingale approach, J. Phys. A: Math. Gen., Vol. 26, 1993, pp. 1823-1834. | MR 1220795 | Zbl 0773.60035
[19] and , One dimensional random Ising system with interaction decay r-(1+∈); a convergent cluster expansion, Commun. Math. Phys., Vol. 111, 1987, pp. 555-557. | MR 901154 | Zbl 0642.60103
[20] , and A. , One dimensional spin glass with interaction decay r-(1+∈); a convergent cluster expansion, Commun. Math. Phys., Vol. 108, 1987, pp. 241-255. | MR 875301
[21] , and , Infinite differentiability for one dimensional spin systems with long range interactions, Commun. Math. Phys., Vol. 87, 1982, pp. 229-252. | MR 684101 | Zbl 0506.58009
[22] , and , On the existence of thermodynamics for the generalised random energy model, J. Stat. Phys., Vol. 46, 1987, pp. 493-505. | MR 883541
[23] , and , Low temperature behaviour of three dimensional spin glasses in a magnetic field, J. Phys. France, Vol. 51, 1990, pp. 1877-1895.
[24] , On the lower critical dimensionality of the Ising model in a random field, J. Phys. C: Sol. State Phys., Vol. 16, 1983, pp. 6615-6622.
[25] and , Boltmann-Gibbs wights in the branching random walk, IMA workshop on branching processes, June 1994, to appear in the proceedings. | MR 1601693 | Zbl 0866.60074
[26] , Spin glasses and other frustrated systems, World Scientific, Singapore, 1986.
[27] , and , The low temperature behaviour of disordered magnets, Commun. Math. Phys., Vol. 100, 1985, pp. 399-437. | MR 802552
[28] , , and , A mean field spin glass with short-range interactions, Commun. Math. Phys., Vol. 106, 1986, pp. 41-89. | MR 853978
[29] and , Spin glass model with random couplings, Commun. Math. Phys., Vol. 93, 1984, pp. 379-406. | MR 745692 | Zbl 0553.60098
[30] and , Large deviations for the multiplicative chaos, Commun. Math. Phys., Vol. 147, 1992, pp. 329-342. | MR 1174416 | Zbl 0755.60022
[31] and , The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: the high temperature case, preprint CMAP École Polytechnique, 1993. | MR 1312435
[32] , A spherical bound for the Sherrington-Kirkpatrick model, preprint Unviersité de Paris VII, 1994. | MR 1417976
[33] , On the strict inequality between quenched and annealed Ising spin glass, preprints SISSA, 1993, to appear in Lett. Math. Phys. | MR 1213613 | Zbl 0768.60096
[34] , Random energy model: limit of a family of disordered models, Phys. Rev. Lett., Vol. 45, 1980, pp. 79-82. | MR 575260
[35] and , Polymers on disordered trees, spin glasses, and travelling waves, J. Stat. Phys., Vol. 51, 1988, pp. 817-840. | MR 971033 | Zbl 1036.82522
[36] , and , Mean-field theory of directed polymers with random complex weights, preprint SPhT, 1992, to appear in Commun. Math. Phys. | MR 1233845
[37] , The description of a random field by means of conditional probabilities and conditions of its regularity, Theor. Prob. Appl., Vol. 13, 1968, pp. 197-224. | MR 231434 | Zbl 0184.40403
[38] and , Random free energies in spin glasses, J. Phys. Lett., Vol. 46, 1985, pp. L909-L914.
[39] , and , Taming Griffiths' singularities: infinite differentiability of quenched correlation functions, preprint UCLA, 1994 to appear in Commun. Math. Phys. | MR 1331689
[40] and , Theory of spin glasses, J. Phys. F: Metal Phys., Vol. 5, 1975, pp. 965-974.
[41] , One-dimensional spin glasses; uniqueness and cluster properties, J. Phys. A: Math. Gen., Vol. 21, 1988, pp. 1781-1786. | MR 952725
[42] , Stiffness exponents, number of pure states, and Almeida-Thouless lines in spin glasses, J. Stat. Phys., Vol. 60, 1990, pp. 275-279. | MR 1063219 | Zbl 01397790
[43] and , Statistical mechanical formalism for sign glasses, Phys. Rev. A, Vol. 29, 1984, pp. 355-365. | MR 729632
[44] and , Thermodynamic limit for long-range spin glasses, J. Stat. Phys., Vol. 32, 1985, pp. 141-152.
[45] and , Absence of symmetry breaking for N-vector spin glass models in two dimensions, Commun. Math. Phys., Vol. 98, 1985, pp. 425-432. | MR 788782
[46] and , The order parameter in a spin glass, Commun. Math. Phys., Vol. 90, 1983, pp. 319-327. | MR 719293
[47] and , Breaking of periodicity at positive temperatures, Commun. Math. Phys., Vol. 134, 1990, pp. 647-651. | MR 1086748 | Zbl 0711.60109
[48] , , and , Continuum limit on a two-dimensional random lattice, Nucl. Phys., Vol. B265 [FS 15], 1986, pp. 92-112.
[49] , and , The Ising model in a random magnetic field, J. Stat. Phys., Vol. 34, 1984, pp. 863-870. | MR 751717
[50] and , Pure phases in spin glasses, J. Phys. A: Math. Gen., Vol. 20, 1987, pp. L997-L1003; Absence of many states in magnetic spin glasses, J. Phys. A: Math. Gen., Vol. 20, 1987, pp. L1005-L1010.
[51] , Mathematical aspects of the physics of disordered systems, in Critical phenomena, random systems, and gauge theories, K OSTERWALDER, R. STORA Eds. pp. 725-894, North-Holland, Amsterdam, 1986. | MR 880538 | Zbl 0669.60098
[52] and , Improved perturbation expansion for disordered systems: beating Griffiths singularities, Commun. Math. Phys., Vol. 96, 1984, pp. 145-180. | MR 768253 | Zbl 0574.60098
[53] and , The disordered phase of long-range Ising spin glasses, Europhys. Lett., Vol. 2, 1986, pp. 53-60.
[54] and , The high temperature phase of long range spin glasses, Commun. Math. Phys., Vol. 110, 1987, pp. 121-155. | MR 885574
[55] and , Some comments on the Sherrington-Kirkpatrick model of spin glasses, Commun. Math. Phys., Vol. 112, 1987, pp. 553-566. | MR 910578
[56] and , Spin glasses and other lattice systems with long range interactions, Commun. Math. Phys., Vol. 120, 1989, pp. 665-688. | MR 987774 | Zbl 0661.60128
[57] , and , Fluctuations in Derrida's random energy and generalised random energy models, J. Stat. Phys., Vol. 54, 1989, pp. 515-529. | MR 984264
[58] , A spin glass on a hierarchical lattice, J. Phys. France, Vol. 45, 1984, pp. 1755-1763. | MR 769601
[59] and , Lattice gas models on self-similar aperiodic tilings, Rev. Math. Phys., Vol. 3, 1991, pp. 163-221. | MR 1121468 | Zbl 0743.60115
[60] , and , Low temperature phase of the Ising spin glass on a hypercubic lattice, Phys. Rev. Lett., Vol. 64, 1990, pp. 2937-2940. | MR 1055998 | Zbl 1050.82558
[61] , , , and , Rigorous bounds for 2D disordered Ising models, J. Phys. France, Vol. 47, 1986, pp. 947-953.
[62] , Spontaneous magnetisation of randomly dilute ferromagnets, J. Stat. Phys., Vol. 25, 1981, pp. 369-396. | MR 630351
[63] , Theory of random determinants, Kluwer Academic Publ., Dodrecht, 1990. | MR 1080966
[64] and , The one-site distribution of Gibbs states on the Bethe lattice are probability vectors of period ≤ 2 for a nonlinear transformation, J. Stat. Phys., Vol. 52, 1988, pp. 267-285. | MR 968586 | Zbl 01397550
[65] , The thermodynamic limit of quenched free energy of magnetic systems with random interactions, J. Phys. A: Math. Gen., Vol. 15, 1982, pp. L51-L54. | MR 638300
[66] , Nonalytic behaviour above the critical point in a random Ising ferromagnet, Phys. Rev. Lett., Vol. 23, 1969, pp. 17-19.
[67] , The stochastic random cluster process and the uniqueness of random cluster measures, preprint Cambridge, 1994.
[68] , Fluctuations and thermodynamic variables in mean field spin glass models, preprint Roma-La Sapienza, 1992.
[69] , The cavity method in the mean field spin glass model. Functional representations of thermodynamic variables, preprint Roma - La Sapienza, 1994. | MR 1414694
[70] , Functional order parameters for the quenched free energy in mean field spin glass model, preprint Roma - La Sapienza, 1992. | MR 1428232
[71] , Classical spin glasses model, Phys. Rev. Lett., Vol. 49, 1982, pp. 409-412. | MR 668161
[72] and Eds., Heidelberg colloquium on spin glasses, Lect. Notes Phys., Vol. 192, 1983. | MR 733796
[73] , Neural networks and physical systems with emergent collective computational capabilities, Proc. Natl. Acad. Sci. USA, Vol. 79, 1982, pp. 2554-2588. | MR 652033
[74] , The ground states of the three-dimensional random field Ising model, Commun. Math. Phys., Vol. 98, 1985, pp. 145-176 | MR 786570
[75] , Random external fields, J. Stat. Phys., Vol. 34, 1984, pp. 849-862.
[76] , Multiplications aléatoires et dimension de Hausdorff, Ann. Inst. H. Poincaré, Vol. 23, 1987, pp. 289-296. | Numdam | MR 898497 | Zbl 0619.60005
[77] , Absence of phase transition in one-dimensional long-range spin systems with random interactions, Theor. Math. Phys., Vol. 43, 1980, pp. 445-449.
[78] and , Existence of free energy for models with long-range random Hamiltonian, J. Stat. Phys., Vol. 20, 1979, pp. 573-584. | MR 537262
[79] and , Infinite-ranged models of spin glasses, Phys. Rev. B, Vol. 17, 1978, pp. 4384-4403.
[80] , A free energy bound for the Hopfield model, J. Phys. A: Math. Gen., Vol. 26, 1993, pp. L353-L355. | MR 1212003 | Zbl 0775.82008
[81] and , Droplet dynamics for asymmetric Ising model, J. Stat. Phys., Vol. 70, 1993, pp. 1121-1148. | MR 1208633 | Zbl 01261826
[82] , , and , Large n limit in statistical mechanics and spectral theory of disordered systems, in Phase transitions and critical phenomena (DOMB and LEBOWITZ Eds.), Vol. 15, 1992, pp. 74-241, Academic Press, London.
[83] , Rigorous bounds for the free energy of short range Ising spin glass model, Europhys. Lett., Vol. 17, 1992, pp. 669-671. | MR 1158147
[84] , A random covering interpretation for the phase transition of the random energy model, J. Stat. Phys., Vol. 60, 1990, pp. 669-674. | MR 1076915 | Zbl 01397779
[85] , private communication, 1994.
[86] , and , Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions, Commun. Math. Phys., Vol. 118, 1988, pp. 365-383. | MR 958802 | Zbl 0668.58013
[87] and , Poisson point processes, cascades, and random coverings of Rn, J. Stat. Phys., Vol. 62, 1991, pp. 481-489. | MR 1105270 | Zbl 0741.60111
[88] and , A rapid dynamical Monte Carlo algorithm for glassy systems, J. Phys. A: Math. Gen., Vol. 27, 1994, pp. L715-L720. | MR 1306164 | Zbl 0850.82045
[89] and , Observable at infinity and states with short range correlations in statistical mechanics, Commun. Math. Phys., Vol. 13, 1969, pp. 194-215. | MR 256687
[90] , Uniqueness of Parisi's scheme for replica symmetry breaking, Ann. Phys. NY, Vol. 216, 1992, pp. 73-97. | MR 1163416 | Zbl 0875.20004
[91] and , Probability in Banach spaces, Springer-Verlag, Berlin, 1991. | MR 1102015 | Zbl 0748.60004
[92] , Pressure and variational principle for random Ising model, Commun. Math. Phys., Vol. 56, 1977, pp. 297-302. | MR 456186 | Zbl 0367.60116
[93] , Combinatorial problems and exercises, North-Holland, Amsterdam, 1993. | MR 1265492 | Zbl 0785.05001
[94] , Critical properties of random spin models from ∈ expansions, Phys. Rev. B, Vol. 11, 1975, pp. 3573-3580.
[95] , The theory of information and coding, Encyclopedia of Mathematics, Addison-Wesley, Reading, 1977. | MR 469516 | Zbl 0376.94011
[96] , and , Replica field theory for determinisitic models: binary sequences with low autocorrelation, preprint Roma-La Sapienza, 1994, available from hep-th@babbage.sissa.it under the reference 9405148.
[97] , and , Replica field theory for determinisitic models: non random spin glasses with glassy behaviour, preprint Roma-La Sapienza, 1994, available from cond-mat@babbage.sissa.it under the reference 9406074.
[98] , , , On the 3d Ising spin, preprint Roma-La Sapienza, 1993, available from cond-mat@babbage.sissa.it under the reference 9310041.
[99] , Dynamical analysis of the low temperature cluster algorithms, J. Stat. phys., Vol. 66, 1992, pp. 1245-1276. | MR 1156404 | Zbl 0925.82181
[100] , and , On the Swendsen-Wang dynamics I: Exponential convergence to the equilibrium, J. Stat. Phys., Vol. 62, 1991, pp. 117-133. | MR 1105259 | Zbl 0739.60097
[101] , and , On the Swendsen-Wang dynamics II: Critical droplets and homogeneous nucleation at low temperature, J. Stat. Phys., Vol. 62, 1991, pp. 135-159. | MR 1105260 | Zbl 0739.60098
[102] and , Brownian bridge polymer model in a random environment, J. Phys. A: Math. Gen., 1996, in press. | MR 1385633
[103] , private communication, 1994.
[104] , and , Spin glass theory and beyond, World Scientific, Singapore, 1987. | MR 1026102 | Zbl 0992.82500
[105] , and , The replica solution without replicas, Europhys. Lett., Vol. 1, 1986, pp. 77-82.
[106] , , , and , Nature of the spin-glass phase, Phys. Rev. Lett., Vol. 52, 1984, pp. 1156-1159.
[107] , A microscopic model with quasicrystalline properties, J. Stat. Phys., Vol. 58, 1990, pp. 1137-1146. | MR 1049061
[108] , Memory capacity in neural network model: rigorous lower bounds, Neural Net., Vol. 1, 1988, pp. 223-238.
[109] and , Multiple states and thermodynamic limites in short ranged Ising spin glass models, Phys. Rev. B, Vol. 46, 1992, pp. 973.
[110] and , Non-mean-field behaviour of realistic spin glasses, preprint New York University, 1995, available from adap-org@xyz.lanl.gov under reference 9508006.
[111] and , Critical behaviour of the three dimensional Ising spin glass model, Phys. Rev. Lett., Vol. 54, 1985, pp. 928-931.
[112] and , On the existence of thermodynamics for the random energy model, Commun. Math. Phys., Vol. 96, 1984, pp. 125-144. | MR 765963 | Zbl 0585.60098
[113] , Infinite number of order parameters for spin glasses, Phys. Rev. Lett., Vol. 43, 1979, pp. 1754-1756.
[114] , A sequence of approximated solutions to the Sherrington-Kirkpatrick model for spin glasses, J. Phys. A: Math. Gen., Vol. 13, 1980, pp. L115-L121.
[115] , The order parameter for spin glasses: a function on the interval [0, 1], J. Phys. A: Math. Gen., Vol. 13, 1980, pp. 1101-1112.
[116] ,, Magnetic properties of spin glasses in a new mean field theory, J. Phys. A: Math. Gen., Vol. 13, 1980, pp. 1887-1895.
[117] , Statistical physics and spectral theory of disordered systems: some recent developments, in Proc. X-th International conference of mathematical physics (Leipzig, 1991), Springer-Verlag, Berlin, 1992. | MR 1386398 | Zbl 0947.82506
[118] and , Theory of disordered spin systems, Theor. Math. Phys., Vol. 35, 1978, pp. 403-414. | MR 503349
[119] and , Spectra of random and almost periodic operators, Springer-Verlag, Berlin, 1992. | MR 1223779 | Zbl 0752.47002
[120] and , Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model, J. Stat. Phys., Vol. 62, 1991, pp. 1-19. | MR 1105253
[121] , and , The replica symmetric solution without replica trick for the Hopfield model, J. Stat. Phys., Vol. 74, 1994, pp. 1161-1183. | MR 1268789 | Zbl 0835.60092
[122] , The zero-level set of the Brownian bridge as a model of disordered physical systems, in Proc. prob. Theory and Math. Statistics Vol. 2 (Vilnius, 1989), Mosklas, Vilnius, 1990. | MR 1153883 | Zbl 0726.60080
[123] , Thermodynamics for the zero-level set of the Brownian bridge, Commun. Math. Phys., Vol. 125, 1989, pp. 579-595. | MR 1024928 | Zbl 0679.60081
[124] , Thermodynamic formalism of neural computing, Lectures given in the IV-th summer school on Statistical Physics and cooperative systems, Santiago de Chile, 12-16 December 1994, preprint Université de Rennes-I, 1995, available from mp-arc@math.utexas.edu under reference 95-221. | MR 1421794
[125] , Simulations numériques Monte Carlo, preprint Université de Rennes-I, 1993, to be published by Masson, Paris, 1995.
[126] , Verres de spin, Thèse d'État, Université de Marseille-II, 1988.
[127] , Rigorous results on spin glasses: a review, Rev. Math. Appl., Vol. 12, 1991, pp. 157-169. | MR 1130608
[128] , Upper bound on the decay of correlations in the plane rotator model with long-range random interaction, J. Stat. Phys., Vol. 36, 1984, pp. 489-516. | MR 780678 | Zbl 0586.60099
[129] , and , Dilute antiferromagnets: Imry-Ma argument, hierarchical model, and equivalence to random field Ising models, J. Stat. Phys., Vol. 75, 1994, pp. 51-65. | MR 1273053 | Zbl 0828.60090
[130] , Random walks in random and non random environments, World Scientific, Singapore, 1990. | MR 1082348 | Zbl 0733.60091
[131] and , Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Phys. Rev., Vol. 96, 1954, pp. 99-102.
[132] , A mathematical reformulation of Derrida's REM and GREM, Commun. Math. Phys., Vol. 108, 1987, pp. 225-239. | MR 875300 | Zbl 0617.60100
[133] and , The free energy of a class of Hopfield models, J. Stat. Phys., Vol. 72, 1993, pp. 113-125. | MR 1233028
[134] , and , Solvable model of a spin glass, Phys. Rev. Lett., Vol. 35, 1975, pp. 1792-1796.
[135] , communication, 1992.
[136] , Concentration of measure and isoperimetric inequalities in product spaces, preprint Université Paris-VI, 1994. | MR 1361756
[137] , A new look at independence, preprint Université de Paris-VI, 1995. | MR 1387624
[138] , Private communication, 1992.
[139] , Thermodynamics of quenched random spin systems and applications to the proble of phase transitions in magnetic spin glass, J. Phys. A: Math. Gen., Vol. 10, 1977, pp. 1319-1333. | MR 452421
[140] , Asymptotic study of a neural network: the Hopfield deterministic sequential dynamics, preprint Université de Rennes-I, 1994.
[141] , Étude asymptotique d'un réseau neuronal, Thèse de doctorat, Université de Rennes-I, 1994.
[142] and , Correlated spin glasses on trees, IMA workshop on branching processes, June 1994, to appear in the proceedings.
[143] , Random fields on random graphs, Adv. Appl. Prob., Vol. 24, 1992, pp. 455-473. | MR 1167267 | Zbl 0753.60102
[144] , Generatingfunctionology, Academic Press, Boston, 1994. | MR 1277813 | Zbl 0831.05001
[145] , Interactions and pressure functionals for disordered lattice systems, Commun. Math. Phys., Vol. 139, 1991, pp. 305-339. | MR 1120141 | Zbl 0747.58066
[146] , Spin glass with long range interactions at high temperature, J. Stat. Phys., Vol. 17, 1987, pp. 911-930. | MR 912509