Communications in Mathematical Physics

, Volume 361, Issue 1, pp 1–52 | Cite as

Spectral Gap Estimates in Mean Field Spin Glasses



We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for reversible nearest-neighbor dynamics and spherical models for Langevin dynamics. Finally, in the case of Ising spins, Panchenko’s recent rigorous calculation (Panchenko in Ann Probab 46(2):865–896, 2018) of the free energy for a system of “two real replica” enables us to prove a quenched LDP for the overlap distribution, which gives us a wider criterion for slow mixing directly related to the Franz–Parisi–Virasoro approach (Franz et al. in J Phys I 2(10):1869–1880, 1992; Kurchan et al. J Phys I 3(8):1819–1838, 1993). This condition holds in a wider range of temperatures.


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  1. 1.
    Aizenman M., Sims R., Starr S.L.: Extended variational principle for the Sherrington-Kirkpatrick spin-glass model. Phys. Rev. B 68(21), 214403 (2003)ADSGoogle Scholar
  2. 2.
    Alon N., Milman V.D.: \({\lambda_1}\), isoperimetric inequalities for graphs, and superconcentrators. J. Comb. Theory Ser. B 38(1), 73–88 (1985)MATHGoogle Scholar
  3. 3.
    Arguin L.-P., Aizenman M.: On the structure of quasi-stationary competing particle systems. Ann. Probab. 37, 1080–1113 (2009)MathSciNetMATHGoogle Scholar
  4. 4.
    Auffinger A., Ben Arous G.: Complexity of random smooth functions on the high-dimensional sphere. Ann. Probab. 41(6), 4214–4247 (2013)MathSciNetMATHGoogle Scholar
  5. 5.
    Auffinger A., Ben Arous G., Černý J.: Random matrices and complexity of spin glasses. Commun. Pure Appl. Math. 66(2), 165–201 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    Auffinger A., Chen W.-K.: On properties of Parisi measures. Probab. Theory Relat. Fields 161(3–4), 817–850 (2015)MathSciNetMATHGoogle Scholar
  7. 7.
    Auffinger A., Chen W.-K.: The Parisi formula has a unique minimizer. Commun. Math. Phys. 335(3), 1429–1444 (2015)ADSMathSciNetMATHGoogle Scholar
  8. 8.
    Auffinger A., Chen W.-K.: Parisi formula for the ground state energy in the mixed p-spin model. Ann. Probab. 45(6b), 4617–4631 (2017)MathSciNetMATHGoogle Scholar
  9. 9.
    Auffinger, A., Chen, W.-K., Zeng, Q.: The SK model is full-step replica symmetry breaking at zero temperature. arXiv preprint arXiv:1703.06872 (2017)
  10. 10.
    Auffinger, A., Jagannath, A.: Thouless–Anderson–Palmer equations for generic p-spin glass models. arXiv preprint arXiv:1612.06359 (2016)
  11. 11.
    Bakry D., Ledoux M.: Lévy–Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 123(2), 259–281 (1996)ADSMathSciNetMATHGoogle Scholar
  12. 12.
    Ben Arous, G.: Aging and spin-glass dynamics. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pp. 3–14. Higher Ed. Press, Beijing (2002)Google Scholar
  13. 13.
    Ben Arous G., Bovier A., Černý J.: Universality of the REM for dynamics of mean-field spin glasses. Commun. Math. Phys. 282(3), 663–695 (2008)ADSMathSciNetMATHGoogle Scholar
  14. 14.
    Ben Arous G., Bovier A., Gayrard V.: Aging in the random energy model. Phys. Rev. Lett. 88(8), 087201 (2002)ADSGoogle Scholar
  15. 15.
    Ben Arous G., Bovier A., Gayrard V.: Glauber dynamics of the random energy model. Commun. Math. Phys. 236(1), 1–54 (2003)ADSMATHGoogle Scholar
  16. 16.
    Ben Arous G., Bovier A., Gayrard V.: Glauber dynamics of the Random Energy Model: II. Aging below the critical temperature. Commun. Math. Phys. 236(1), 1–54 (2003)ADSMathSciNetMATHGoogle Scholar
  17. 17.
    Ben Arous G., Gün O.: Universality and extremal aging for dynamics of spin glasses on subexponential time scales. Commun. Pure Appl. Math. 65(1), 77–127 (2012)MathSciNetMATHGoogle Scholar
  18. 18.
    Berthier L., Biroli G.: Theoretical perspective on the glass transition and amorphous materials. Rev. Mod. Phys. 83(2), 587 (2011)ADSGoogle Scholar
  19. 19.
    Biroli G.: Dynamical tap approach to mean field glassy systems. J. Phys. A Math. Gen. 32(48), 8365 (1999)ADSMathSciNetMATHGoogle Scholar
  20. 20.
    Bouchaud J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I 2(9), 1705–1713 (1992)Google Scholar
  21. 21.
    Bouchaud, J.-P., Cugliandolo, L.F., Kurchan, J., Mézard, M.: Out of equilibrium dynamics in spin-glasses and other glassy systems. In: Young, A.P. (ed.) Spin Glasses and Random Fields, Series on Directions in Condensed Matter Physics, vol. 12, pp. 161–223 (1998)Google Scholar
  22. 22.
    Bouchaud J.-P., Dean D.S.: Aging on parisi’s tree. J. Phys. I 5(3), 265–286 (1995)Google Scholar
  23. 23.
    Bovier A., Faggionato A.: Spectral characterization of aging: the REM-like trap model. Ann. Appl. Probab. 15(3), 1997–2037 (2005)MathSciNetMATHGoogle Scholar
  24. 24.
    Bovier A., Gayrard V.: Convergence of clock processes in random environments and ageing in the p-spin SK model. Ann. Probab. 41(2), 817–847 (2013)MathSciNetMATHGoogle Scholar
  25. 25.
    Bovier A., Klimovsky A.: The Aizenman-Sims-Starr and Guerra’s schemes for the SK model with multidimensional spins. Electronic Journal of Probability 14(8), 161–241 (2009)MathSciNetMATHGoogle Scholar
  26. 26.
    Buser P.: A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15(2), 213–230 (1982)MathSciNetMATHGoogle Scholar
  27. 27.
    Castellani T., Cavagna A.: Spin-glass theory for pedestrians. J. Stat. Mech. Theory Exp. 2005(05), P05012 (2005)MathSciNetGoogle Scholar
  28. 28.
    Černý J., Wassmer T.: Aging of the Metropolis dynamics on the random energy model. Probab. Theory Relat. Fields 167(1-2), 253–303 (2017)MathSciNetMATHGoogle Scholar
  29. 29.
    Chatterjee, S.: The Ghirlanda–Guerra identities without averaging. arXiv preprint arXiv:0911.4520 (2009)
  30. 30.
    Chavel I.: Eigenvalues in Riemannian Geometry, Volume 115 of Pure and Applied Mathematics. Academic Press, Inc., Orlando (1984) Including a chapter by Burton Randol, With an appendix by Jozef DodziukGoogle Scholar
  31. 31.
    Cheeger, J.: A lower bound for the smallest eigenvalue of the laplacian. In: Proceedings of the Princeton conference in honor of Professor S. Bochner (1969)Google Scholar
  32. 32.
    Chen, W.K.: Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound. Ann. Probab. 45(6A), 3929–3966 (2017)Google Scholar
  33. 33.
    Chen W.-K.: The Aizenman–Sims–Starr scheme and Parisi formula for mixed p-spin spherical models. Electron. J. Probab. 18(94), 14 (2013)MathSciNetMATHGoogle Scholar
  34. 34.
    Chen W.-K., Dey P., Panchenko D.: Fluctuations of the free energy in the mixed p-spin models with external field. Probab. Theory Relat. Fields 168, 1–13 (2015)MathSciNetGoogle Scholar
  35. 35.
    Chen, W.K., Handschy, M., Lerman, G.: On the energy landscape of the mixed even p-spin model. Probab. Theory Relat. Fields 171, 53 (2018)Google Scholar
  36. 36.
    Chen W.-K., Hsieh H.-W., Hwang C.-R., Sheu Y.-C.: Disorder chaos in the spherical mean-field model. J. Stat. Phys. 160(2), 417–429 (2015)ADSMathSciNetMATHGoogle Scholar
  37. 37.
    Cugliandolo, L.F.: Course 7: Dynamics of glassy systems. In: Barrat, J.-L., Feigelman, M.V., Kurchan, J., Dalibard, J. (eds.) Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, Les Houches Session LXXVII, pp. 367–521. Springer, Berlin (2002)Google Scholar
  38. 38.
    Almeida J.R.L., Thouless D.J.: Stability of the Sherrington–Kirkpatrick solution of a spin glass model. J. Phys. A Math. Gen. 11(5), 983 (1978)ADSGoogle Scholar
  39. 39.
    Santis E.: Glauber dynamics of spin glasses at low and high temperature. Ann. Inst. H. Poincaré Probab. Stat. 38(5), 681–710 (2002)ADSMathSciNetMATHGoogle Scholar
  40. 40.
    Diaconis P., Saloff-Coste L. et al.: Logarithmic sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6(3), 695–750 (1996)MathSciNetMATHGoogle Scholar
  41. 41.
    Evans L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd ed. American Mathematical Society, Providence (2010)Google Scholar
  42. 42.
    Fontes L.R., Isopi M., Kohayakawa Y., Picco P.: The spectral gap of the REM under Metropolis dynamics. Ann. Appl. Probab. 8(3), 917–943 (1998)MathSciNetMATHGoogle Scholar
  43. 43.
    Franz S., Parisi G., Virasoro M.A.: The replica method on and off equilibrium. J. Phys. I 2(10), 1869–1880 (1992)Google Scholar
  44. 44.
    Gayrard, V.: Aging in metropolis dynamics of the REM: a proof. arXiv preprint arXiv:1602.06081 (2016)
  45. 45.
    Gayrard V.: Convergence of clock processes and aging in Metropolis dynamics of a truncated REM. Ann. Henri Poincaré 17(3), 537–614 (2016)ADSMathSciNetMATHGoogle Scholar
  46. 46.
    Gheissari, R., Jagannath, A.: On the spectral gap of spherical spin glass dynamics. Ann. Henri Poincare Probab. Stat. arXiv preprint arXiv:1608.06609 (2016)
  47. 47.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001) Reprint of the 1998 editionMATHGoogle Scholar
  48. 48.
    Guerra, F.: Sum rules for the free energy in the mean field spin glass model. In: Longo, R. (ed.) Mathematical Physics in Mathematics and Physics (Siena, 2000), Volume 30 of Fields Inst. Commun., pp. 161–170. Amer. Math. Soc., Providence (2001)Google Scholar
  49. 49.
    Guerra F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003)ADSMathSciNetMATHGoogle Scholar
  50. 50.
    Guerra F., Toninelli F.L.: The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230(1), 71–79 (2002)ADSMathSciNetMATHGoogle Scholar
  51. 51.
    Guionnet, A.: Dynamics for spherical models of spin-glass and aging. In: Bovier, A., Bolthausen, E. (eds.) Spin Glasses, pp. 117–144. Springer, Berlin (2007)Google Scholar
  52. 52.
    Guionnet A., Zegarlinski B.: Decay to equilibrium in random spin systems on a lattice. Commun. Math. Phys. 181(3), 703–732 (1996)ADSMathSciNetMATHGoogle Scholar
  53. 53.
    Guionnet, A., Zegarlinski, B.: Lectures on logarithmic Sobolev inequalities. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds.) Séminaire de Probabilités, XXXVI, Volume 1801 of Lecture Notes in Math., pp. 1–134. Springer, Berlin (2003)Google Scholar
  54. 54.
    Holley R., Stroock D.: Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46(5-6), 1159–1194 (1987)ADSMathSciNetMATHGoogle Scholar
  55. 55.
    Jagannath A., Tobasco I.: Bounding the complexity of replica symmetry breaking for spherical spin glasses. Proc. Am. Math. Soc. 146(7), 3127–3142 (2016)MATHGoogle Scholar
  56. 56.
    Jagannath A., Tobasco I.: A dynamic programming approach to the Parisi functional. Proc. Am. Math. Soc. 144(7), 3135–3150 (2016)MathSciNetMATHGoogle Scholar
  57. 57.
    Jagannath A., Tobasco I.: Low temperature asymptotics of spherical mean field spin glasses. Commun. Math. Phys. 352(3), 979–1017 (2017)ADSMathSciNetMATHGoogle Scholar
  58. 58.
    Jagannath A., Tobasco I.: Some properties of the phase diagram for mixed p-spin glasses. Probab. Theory Relat. Fields 167(3–4), 615–672 (2017)MathSciNetMATHGoogle Scholar
  59. 59.
    Jerrum M., Sinclair A.: Approximating the permanent. SIAM J Comput 18(6), 1149–1178 (1989)MathSciNetMATHGoogle Scholar
  60. 60.
    Kurchan J., Parisi G., Virasoro M.A.: Barriers and metastable states as saddle points in the replica approach. J. Phys. I 3(8), 1819–1838 (1993)Google Scholar
  61. 61.
    Lawler G.F., Sokal A.D.: Bounds on the L 2 spectrum for markov chains and markov processes: a generalization of cheeger’s inequality. Trans. Am. Math. Soc. 309(2), 557–580 (1988)MATHGoogle Scholar
  62. 62.
    Lax P.D.: Functional Analysis. Pure and Applied Mathematics (New York). Wiley-Interscience, New York (2002)Google Scholar
  63. 63.
    Ledoux M.: A simple analytic proof of an inequality by P. Buser. Proc. Am. Math. Soc. 121(3), 951–959 (1994)MathSciNetMATHGoogle Scholar
  64. 64.
    Ledoux M.: The Concentration of Measure Phenomenon, Volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001)Google Scholar
  65. 65.
    Ledoux M., Talagrand M.: Probability in Banach Spaces. Classics in Mathematics. Springer, Berlin (2011) Isoperimetry and processes, Reprint of the 1991 editionMATHGoogle Scholar
  66. 66.
    Levin D.A., Peres Y., Wilmer E.L.: Markov Chains and Mixing Times. American Mathematical Soc., Providence (2009)MATHGoogle Scholar
  67. 67.
    Mathieu P.: Convergence to equilibrium for spin glasses. Commun. Math. Phys. 215(1), 57–68 (2000)ADSMathSciNetMATHGoogle Scholar
  68. 68.
    Mathieu P., Mourrat J.-C.: Aging of asymmetric dynamics on the random energy model. Probab. Theory Relat. Fields 161(1–2), 351–427 (2015)MathSciNetMATHGoogle Scholar
  69. 69.
    Mélin R., Butaud P.: Glauber dynamics and ageing. J. Phys. I 7(5), 691–710 (1997)MathSciNetGoogle Scholar
  70. 70.
    Mézard M., Parisi G., Virasoro M.A.: Spin Glass Theory and Beyond, vol. 9. World scientific Singapore, (1987)Google Scholar
  71. 71.
    Montanari A., Semerjian G.: Rigorous inequalities between length and time scales in glassy systems. J. Stat. Phys. 125(1), 23 (2006)ADSMathSciNetMATHGoogle Scholar
  72. 72.
    Panchenko D.: A note on Talagrand’s positivity principle. Electron. Commun. Probab. 12, 401–410 (2007)MathSciNetMATHGoogle Scholar
  73. 73.
    Panchenko D.: On differentiability of the Parisi formula. Electron. Commun. Probab. 13, 241–247 (2008)MathSciNetMATHGoogle Scholar
  74. 74.
    Panchenko D.: The Ghirlanda–Guerra identities for mixed p-spin model. C. R. Math. Acad. Sci. Paris 348(3–4), 189–192 (2010)MathSciNetMATHGoogle Scholar
  75. 75.
    Panchenko D.: The Parisi ultrametricity conjecture. Ann. Math. (2) 177(1), 383–393 (2013)MathSciNetMATHGoogle Scholar
  76. 76.
    Panchenko D.: The Sherrington–Kirkpatrick Model. Springer, Berlin (2013)MATHGoogle Scholar
  77. 77.
    Panchenko D.: The Parisi formula for mixed p-spin models. Ann. Probab. 42(3), 946–958 (2014)MathSciNetMATHGoogle Scholar
  78. 78.
    Panchenko D.: Chaos in temperature in generic 2p-spin models. Commun. Math. Phys. 346(2), 703–739 (2016)ADSMathSciNetMATHGoogle Scholar
  79. 79.
    Panchenko D.: Free energy in the mixed p-spin models with vector spins. Ann. Probab. 46(2), 865–896 (2018)MathSciNetMATHGoogle Scholar
  80. 80.
    Panchenko D., Talagrand M.: On the overlap in the multiple spherical SK models. Ann. Probab. 35(6), 2321–2355 (2007)MathSciNetMATHGoogle Scholar
  81. 81.
    Parisi G.: A sequence of approximated solutions to the SK model for spin glasses. J. Phys. A Math. Gen. 13(4), L115 (1980)ADSGoogle Scholar
  82. 82.
    Parisi G.: Order parameter for spin-glasses. Phys. Rev. Lett. 50(24), 1946 (1983)ADSMathSciNetGoogle Scholar
  83. 83.
    Saloff-Coste, L.: Lectures on finite Markov chains. In: Bernard, P. (ed.) Lectures on probability theory and statistics (Saint-Flour, 1996), Volume 1665 of Lecture Notes in Math., pp. 301–413. Springer, Berlin (1997)Google Scholar
  84. 84.
    Stroock D.W., Srinivasa Varadhan S.R.: Multidimensional Diffussion Processes, vol. 233. Springer, Berlin (1979)Google Scholar
  85. 85.
    Subag E.: The complexity of spherical p-spin models: a second moment approach. Ann. Probab. 45(5), 3385–3450 (2017)MathSciNetMATHGoogle Scholar
  86. 86.
    Subag E.: The geometry of the gibbs measure of pure spherical spin glasses. Invent. Math. 210(1), 135–209 (2017)ADSMathSciNetMATHGoogle Scholar
  87. 87.
    Subag E., Zeitouni O.: The extremal process of critical points of the pure p-spin spherical spin glass model. Probab. Theory Relat. Fields 168(3-4), 773–820 (2016)MathSciNetMATHGoogle Scholar
  88. 88.
    Talagrand M.: On Guerra’s broken replica-symmetry bound. C. R. Math. Acad. Sci. Paris 337(7), 477–480 (2003)MathSciNetMATHGoogle Scholar
  89. 89.
    Talagrand M.: Free energy of the spherical mean field model. Probab. Theory Relat. Fields 134(3), 339–382 (2006)MathSciNetMATHGoogle Scholar
  90. 90.
    Talagrand M.: Parisi measures. J. Funct. Anal. 231(2), 269–286 (2006)MathSciNetMATHGoogle Scholar
  91. 91.
    Talagrand M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)MathSciNetMATHGoogle Scholar
  92. 92.
    Talagrand, M.: Mean Field Models for Spin Glasses. Volume I, Volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (2011). Basic examplesGoogle Scholar
  93. 93.
    Talagrand, M.: Mean field models for spin glasses. Volume II, Volume 55 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg (2011). Advanced replica-symmetry and low temperatureGoogle Scholar

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Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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