Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 515–542 | Cite as

The Marginally Stable Bethe Lattice Spin Glass Revisited

  • Giorgio Parisi


Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a spin glass on a Bethe lattice has been studied only using an approximation where marginal stability is not present, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a marginally stable solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non-perturbative approach to the Bethe lattice spin glass problem using approximations that should be hopefully consistent with marginal stability.


Spin glasses Bethe lattices Replicas 



It is a pleasure to thank Dmitry Panchenko for many discussions and clarifications. I would like also to thank Valerio Astuti, Silvio Franz, Carlo Lucibello, Federico Ricci-Tersenghi, and Pierfrancesco Urbani for a critical reading of the manuscript. This work has been supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant Agreement No. [694925]).


  1. 1.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987)zbMATHGoogle Scholar
  2. 2.
    Parisi, G.: Field Theory, Disorder and Simulations. World Scientific, Singapore (1992)CrossRefzbMATHGoogle Scholar
  3. 3.
    Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792 (1975)ADSCrossRefGoogle Scholar
  4. 4.
    De Dominicis, C., Kondor, I.: Eigenvalues of the stability matrix for Parisi solution of the long-range spin-glass. Phys. Rev. B 27, 606 (1983)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233, 1 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Talagrand, M.: The generalized Parisi formula. C. R. Math. 337, 111 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Talagrand, M.: The Parisi formula. Ann. Math. 163, 221 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Talagrand, M.: Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models. Springer, Berlin (2003)zbMATHGoogle Scholar
  9. 9.
    Panchenko, D.: The Sherrington–Kirkpatrick Model. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Panchenko, D.: The Sherrington–Kirkpatrick model: an overview. J. Stat. Phys. 149, 362 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Panchenko, D.: Introduction to the SK model. arXiv:1412.0170 (2014)
  12. 12.
    Contucci, P., Giardinà, C.: Perspectives on Spin Glasses. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  13. 13.
    Kirkpatrick, T.R., Thirumalai, D.: Dynamics of the structural glass transition and the p-spin—interaction spin-glass model. Phys. Rev. Lett. 58, 2091 (1987)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kirkpatrick, T.R., Thirumalai, D.: p-spin-interaction spin-glass models: connections with the structural glass problem. Phys. Rev. B 36, 5388 (1987)ADSCrossRefGoogle Scholar
  15. 15.
    Kirkpatrick, T.R., Thirumalai, D., Wolynes, P.G.: Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. Phys. Rev. A 40, 1045 (1989)ADSCrossRefGoogle Scholar
  16. 16.
    Panchenko, D.: Free energy in the Potts spin glass. arXiv:1512.00370 (2015)
  17. 17.
    Panchenko, D.: Free energy in the mixed p-spin models with vector spins. arXiv:1512.04441 (2015)
  18. 18.
    Panchenko, D.: On the K-sat model with large number of clauses. arXiv:1608.06256 (2016)
  19. 19.
    Barra, A., Contucci, P., Mingione, E., Tantari, D.: Multi-species mean field spin glasses. Rigorous results. In: Annales Henri Poincaré, vol. 16, p. 691. Springer, Berlin (2015)Google Scholar
  20. 20.
    Panchenko, D.: The free energy in a multi-species Sherrington–Kirkpatrick model. Ann. Prob. 43, 3494 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Panchenko, D.: Chaos in temperature in generic 2p-spin models. Commun. Math. Phys. 346, 703 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chen, W.-K., Panchenko, D.: Temperature chaos in some spherical mixed p-spin models. arXiv:1608.02478 (2016)
  23. 23.
    De Dominicis, C., Kondor, I.: On spin glass fluctuations. J. Phys. Lett. 45, 205 (1984)CrossRefGoogle Scholar
  24. 24.
    Alvarez Banos, R., Cruz, A., Fernandez, L.A., Gil-Narvion, J.M., Gordillo-Guerrero, A., Guidetti, M., Maiorano, A., Mantovani, F., Marinari, E., Martin-Mayor, V., Monforte-Garcia, J., Munoz Sudupe, A., Navarro, D., Parisi, G., Perez-Gaviro, S., Ruiz-Lorenzo, J.J., Schifano, S.F., Seoane, B., Tarancon, A., Tripiccione, R., Yllanes, D.: Static versus dynamic heterogeneities in the D= 3 Edwards-Anderson-Ising spin glass. Phys. Rev. Lett. 105, 177202 (2010)ADSCrossRefGoogle Scholar
  25. 25.
    Alvarez Banos, R., Cruz, A., Fernandez, L.A., Gil-Narvion, J.M., Gordillo-Guerrero, A., Guidetti, M., Maiorano, A., Mantovani, F., Marinari, E., Martin-Mayor, V., Monforte-Garcia, J., Munoz Sudupe, A., Navarro, D., Parisi, G., Perez-Gaviro, S., Ruiz-Lorenzo, J.J., Schifano, S.F., Seoane, B., Tarancon, A., Tripiccione, R., Yllanes, D.: Nature of the spin-glass phase at experimental length scales. J. Stat. Mech. P06026 (2010)Google Scholar
  26. 26.
    Wyart, M., Silbert, L.E., Nagel, S.R., Witten, T.A.: Effects of compression on the vibrational modes of marginally jammed solids. Phys. Rev. E 72, 051306 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    DeGiuli, E., Laversanne-Finot, A., Düring, G.A., Lerner, E., Wyart, M.: Effects of coordination and pressure on sound attenuation, boson peak and elasticity in amorphous solids. Soft Matter 10, 5628 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    Marinari, E., Parisi, G., Ricci-Tersenghi, F., Ruiz-Lorenzo, J., Zuliani, F.: Replica symmetry breaking in short-range spin glasses: theoretical foundations and numerical evidences. J. Stat. Phys. 98, 973 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kurchan, J., Parisi, G., Zamponi, F.: Exact theory of dense amorphous hard spheres in high dimension I. The free energy. J. Stat. Mech. P10012 (2012)Google Scholar
  30. 30.
    Kurchan, J., Parisi, G., Urbani, P., Zamponi, F.: Exact theory of dense amorphous hard spheres in high dimension. II. The high density regime and the Gardner transition. J. Phys. Chem. B 117, 12979 (2013)CrossRefGoogle Scholar
  31. 31.
    Charbonneau, P., Kurchan, J., Parisi, G., Urbani, P., Zamponi, F.: Fractal free energy landscapes in structural glasses. Nat. Commun. 5, 3725 (2014)ADSCrossRefGoogle Scholar
  32. 32.
    Charbonneau, P., Kurchan, J., Parisi, G., Urbani, P., Zamponi, F.: Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution. J. Stat. Mech. P10009 (2014)Google Scholar
  33. 33.
    Charbonneau, P., Kurchan, J., Parisi, G., Urbani, P., Zamponi, F.: Glass and jamming transitions: from exact results to finite-dimensional descriptions arXiv:1605.03008 (2016)
  34. 34.
    Ruelle, D.: A mathematical reformulation of Derrida’s REM and GREM. Commun. Math. Phys. 108, 225 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bolthausen, E., Sznitman, A.S.: On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197, 247 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Panchenko, D.: Spin glass models from the point of view of spin distributions. Ann. Prob. 41, 1315 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Panchenko, D.: Hierarchical exchangeability of pure states in mean field spin glass models. Prob. Theory Relat. Fields 161, 619 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111, 535 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Franz, S., Leone, M., Toninelli, F.L.: Replica bounds for diluted non-Poissonian spin systems. J. Phys. A 43, 10967 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Aizenman, M., Sims, R., Starr, S.L.: Extended variational principle for the Sherrington–Kirkpatrick spin-glass model. Phys. Rev. B 68, 214403 (2003)ADSCrossRefGoogle Scholar
  41. 41.
    Panchenko, D.: The Parisi ultrametricity conjecture. Ann. Math. 177, 383 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Panchenko, D.: Structure of finite-RSB asymptotic Gibbs measures in the diluted spin glass models. J. Stat. Phys. 162, 1 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Mézard, M., Parisi, G.: The Bethe lattice spin glass revisited. Eur. Phys. J. B 20, 217 (2001)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Viana, L., Bray, A.J.: Phase diagrams for dilute spin glasses. J. Phys. C 18, 3037 (1985)ADSCrossRefGoogle Scholar
  45. 45.
    Fu, Y.T., Anderson, P.W.: Application of statistical mechanics to NP-complete problems in combinatorial optimisation. J. Phys. A 19, 1065 (1986)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Mézard, M., Parisi, G.: Mean-field theory of randomly frustrated systems with finite connectivity. EPL 3, 1067 (1987)ADSCrossRefGoogle Scholar
  47. 47.
    De Dominicis, C.: Replica symmetry breaking in finite connectivity systems: a large connectivity expansion at finite and zero temperature. J. Phys. A 22, L775 (1989)ADSCrossRefGoogle Scholar
  48. 48.
    Parisi, G.: Glasses, replicas and all that. In: Barrat, J.L. et al. (eds.) Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter: Les Houches Session LXXVII. Springer, Berlin (2005)Google Scholar
  49. 49.
    Morone, F., Caltagirone, F., Harrison, E., Parisi, G.: Replica theory and spin glasses. arXiv:1409.2722 (2014)
  50. 50.
    Guerra, F.: About the overlap distribution in mean field spin glass models. Int. J. Mod. Phys. B 10, 1675 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Ghirlanda, S., Guerra, F.: General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31, 9149 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Aizenman, M., Contucci, P.: On the stability of the quenched state in mean-field spin-glass models. J. Stat. Phys. 92, 765 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Parisi, G.: On the probability distribution of the overlap in spin glasses. Int. J. Mod. Phys. B 18, 733 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Parisi, G., Ricci-Tersenghi, F.: On the origin of ultrametricity. J. Phys. A 33, 113 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Parisi, G.: On the branching structure of the tree of states in spin glasses. J. Stat. Phys. 72, 857 (1993)ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Mézard, M., Parisi, G.: The cavity method at zero temperature. J. Stat. Phys. 111, 1 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812 (2002)ADSCrossRefGoogle Scholar
  58. 58.
    Mézard, M., Virasoro, M.A.: The microstructure of ultrametricity. J. Phys. 46, 1293 (1985)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Goldschmidt, C., Martin, J.: Random recursive trees and the Bolthausen-Sznitman coalescent. E. J. Prob. 10, 718 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Ruzmaikina, A., Aizenman, M.: Characterization of invariant measures at the leading edge for competing particle systems. Ann. Prob. 33, 82 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Parisi, G.: On spin glass theory. Phys. Scr. T19A, 27 (1987)ADSCrossRefGoogle Scholar
  62. 62.
    Parisi, G., Ricci-Tersenghi, F., Yllanes, D.: Explicit generation of the branching tree of states in spin glasses. J. Stat. Mech. P05002 (2015)Google Scholar
  63. 63.
    Mézard, M., Parisi, G., Virasoro, M.A.: Random free energies in spin glasses. J. Phys. Lett. 46, 217 (1985)CrossRefGoogle Scholar
  64. 64.
    Guerra, F., Tolinelli, F.L.: The high temperature region of the Viana–Bray diluted spin glass model. J. Stat. Phys. 115, 531 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Sen, S.: Optimization on sparse random hypergraphs and spin glasses. arXiv:1606.02365 (2016)
  66. 66.
    Auffinger, A., Chen, W.-K.: The Parisi formula has a unique minimizer. Commun. Math. Phys. 335, 1429 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Parisi, G.: Spin glasses. In: Cabibbo, N. (ed.) Proceedings of the International School of Physics ’Enrico Fermi’ Course XCII: Elementary Particles. North-Holland, Amsterdam (1984)Google Scholar
  68. 68.
    Sommers, H.-J., Dupont, W.: Distribution of frozen fields in the mean-field theory of spin glasses. J. Phys. C 7, 5785 (1984)ADSCrossRefGoogle Scholar
  69. 69.
    Crisanti, A., Rizzo, T.: Analysis of the \( \infty \)-replica symmetry breaking solution of the Sherrington–Kirkpatrick model. Phys. Rev. E 65, 046137 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Parisi, G., Rizzo, T.: Large deviations of the free energy in diluted mean-field spin-glass. J. Phys. A 43, 045001 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Parisi, G., Rizzo, T.: Chaos in temperature in diluted mean-field spin-glass. J. Phys. A 43, 235003 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Parisi, G., Tria, F.: Spin glasses on Bethe lattices for large coordination number. Eur. Phys. J. B 30, 533 (2002)ADSCrossRefGoogle Scholar
  73. 73.
    Boschi, G.: Rome University La Sapienza thesis, unpublished (2016)Google Scholar
  74. 74.
    Morone, F., Parisi, G., Ricci-Tersenghi, F.: Large deviations of correlation functions in random magnets. Phys. Rev. B 89, 214202 (2014)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Dipartimento di Fisica, INFN, Sezione di Roma I, CNR-NANOTEC UOS RomaUniversità di Roma La SapienzaRomaItaly

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