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Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

  • Erik BatesEmail author
  • Leila Sloman
  • Youngtak Sohn
Article

Abstract

In the Sherrington–Kirkpatrick (SK) and related mixed p-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first non-asymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.

Keywords

Spin glasses Sherrington–Kirkpatrick model de Almeida–Thouless line 

Mathematics Subject Classification

60K35 82B26 82B44 

Notes

Acknowledgements

We are grateful to Amir Dembo and Andrea Montanari for their advice and encouragement on this project. We thank Antonio Auffinger, Erwin Bolthausen, and Aukosh Jagannath for their insights and feedback, and the referee for several suggestions to improve the manuscript.

References

  1. 1.
    Agliari, E., Barra, A., Galluzzi, A., Guerra, F., Moauro, F.: Multitasking associative networks. Phys. Rev. Lett. 109, 1–5 (2012)CrossRefGoogle Scholar
  2. 2.
    Agliari, E., Barra, A., Guerra, F., Moauro, F., Agliari, E., Barra, A., Guerra, F., Moauro, F.: A thermodynamic perspective of immune capabilities. J. Theor. Biol. 287, 48–63 (2011)CrossRefGoogle Scholar
  3. 3.
    Aizenman, M., Lebowitz, J.L., Ruelle, D.: Some rigorous results on the Sherrington–Kirkpatrick spin glass model. Commun. Math. Phys. 112(1), 3–20 (1987)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Amit, D.J., Gutfreund, H., Sompolinsky, H.: Spin-glass models of neural networks. Phys. Rev. A 32(2), 1007–1018 (1985)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Auffinger, A., Chen, W.-K.: On properties of Parisi measures. Probab. Theory Relat. Fields 161(3–4), 817–850 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Auffinger, A., Chen, W.-K.: The Parisi formula has a unique minimizer. Commun. Math. Phys. 335(3), 1429–1444 (2015)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Auffinger, A., Chen, W.-K., Zeng, Q.: The SK model is full-step replica symmetry breaking at zero temperature. Preprint, available at arXiv:1703.06872
  8. 8.
    Barra, A., Agliari, E.: A statistical mechanics approach to autopoietic immune networks. J. Stat Mech. Theory E 2010(07), 1–24 (2010)CrossRefGoogle Scholar
  9. 9.
    Barra, A., Agliari, E.: A statistical mechanics approach to Granovetter theory. Phys. A 391(10), 3017–3026 (2012)CrossRefGoogle Scholar
  10. 10.
    Barra, A., Contucci, P.: Toward a quantitative approach to migrants integration. Europhys. Lett. 89(6), 1–6 (2010)CrossRefGoogle Scholar
  11. 11.
    Barra, A., Contucci, P., Mingione, E., Tantari, D.: Multi-species mean field spin glasses. Rigorous results. Ann. Henri Poincaré 16(3), 691–708 (2015)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Barra, A., Galluzzi, A., Pizzoferrato, A., Tantari, D.: Mean field bipartite spin models treated with mechanical techniques. Eur. Phys. J. 87(3), 74 (2014)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Barra, A., Genovese, G., Guerra, F.: The replica symmetric approximation of the analogical neural network. J. Stat. Phys. 140(4), 784–796 (2010)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Barra, A., Genovese, G., Guerra, F.: Equilibrium statistical mechanics of bipartite spin systems. J. Phys. A 44(24), 245002, 22 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Barra, A., Genovese, G., Guerra, F., Tantari, D.: How glassy are neural networks? J. Stat. Mech. 07, 1–16 (2012)Google Scholar
  16. 16.
    Barra, A., Guerra, F.: About the ergodic regime in the analogical Hopfield neural networks: moments of the partition function. J. Math. Phys. 49(12), 1–18 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bovier, A., Picco, P. (eds.): Mathematical Aspects of Spin Glasses and Neural Networks. Progress in Probability, vol. 41. Birkhäuser Boston Inc, Boston, MA (1998)Google Scholar
  18. 18.
    Contucci, P., Gallo, I., Menconi, G.: Phase transitions in social sciences: two-population mean field theory. Int. J. Mod. Phys. B 22(14), 2199–2212 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    Contucci, P., Ghirlanda, S.: Modeling society with statistical mechanics: an application to cultural contact and immigration. Qual. Quant. 41(4), 569–578 (2007)CrossRefGoogle Scholar
  20. 20.
    de Almeida, J.R.L., Thouless, D.J.: Stability of the Sherrington–Kirkpatrick solution of a spin glass model. J. Phys. A 11(5), 983–990 (1978)ADSCrossRefGoogle Scholar
  21. 21.
    Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F 5(5), 965–974 (1975)ADSCrossRefGoogle Scholar
  22. 22.
    Fedele, M., Contucci, P.: Scaling limits for multi-species statistical mechanics mean-field models. J. Stat. Phys. 144(6), 1186–1205 (2011)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Fedele, M., Unguendoli, F.: Rigorous results on the bipartite mean-field model. J. Phys. A 45(38), 385001, 18 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gallo, I., Contucci, P.: Bipartite mean field spin systems. Existence and solution. Math. Phys. Electron. J. 14 (2008), Paper 1, 21Google Scholar
  25. 25.
    Guerra, F.: Sum, rules for the free energy in the mean field spin glass model. In: Mathematical physics in mathematics and physics (Siena, : vol. 30 of Fields Institute Communications American Mathematical Society. Providence, RI, 2001), 161–170 (2000)Google Scholar
  26. 26.
    Guerra, F., Toninelli, F.L.: Quadratic replica coupling in the Sherrington–Kirkpatrick mean field spin glass model. J. Math. Phys. 43(7), 3704–3716 (2002)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230(1), 71–79 (2002)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Hopfield, J.J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79(8), 2554–2558 (1982)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Krapivsky, P.L., Redner, S.: Dynamics of majority rule in two-state interacting spin systems. Phys. Rev. Lett. 90, 1–4 (2003)CrossRefGoogle Scholar
  31. 31.
    Latała, R.: Exponential inequalities for the SK model of spin glasses, extending Guerras method. Unpublished manuscript (2002)Google Scholar
  32. 32.
    Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford Graduate Texts. Oxford University Press, Oxford (2009)CrossRefGoogle Scholar
  33. 33.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and beyond. World Scientific Lecture Notes in Physics, vol. 9. World Scientific Publishing Co. Inc., Teaneck (1987)zbMATHGoogle Scholar
  34. 34.
    Nishimori, H.: Statistical physics of spin glasses and information processing, vol. 111 of International Series of Monographs on Physics. Oxford University Press, New York, 2001. An introduction, Translated from the 1999 Japanese originalGoogle Scholar
  35. 35.
    Panchenko, D.: Free energy in the generalized Sherrington–Kirkpatrick mean field model. Rev. Math. Phys. 17(7), 793–857 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Panchenko, D.: The Sherrington–Kirkpatrick Model. Springer Monographs in Mathematics. Springer, New York (2013)CrossRefGoogle Scholar
  37. 37.
    Panchenko, D.: The Parisi formula for mixed \(p\)-spin models. Ann. Probab. 42(3), 946–958 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Panchenko, D.: The free energy in a multi-species Sherrington–Kirkpatrick model. Ann. Probab. 43(6), 3494–3513 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Parisi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754–1756 (1979)ADSCrossRefGoogle Scholar
  40. 40.
    Parisi, G.: A sequence of approximated solutions to the S–K model for spin glasses. J. Phys. A 13(4), L115–L121 (1980)ADSCrossRefGoogle Scholar
  41. 41.
    Parisi, G.: A simple model for the immune network. Proc. Natl. Acad. Sci. USA 87(1), 429–433 (1990)ADSCrossRefGoogle Scholar
  42. 42.
    Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35(26), 1792–1796 (1975)ADSCrossRefGoogle Scholar
  43. 43.
    Talagrand, M.: On the high temperature phase of the Sherrington–Kirkpatrick model. Ann. Probab. 30(1), 364–381 (2002)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Talagrand, M.: The Parisi formula. Ann. Math. 163(2), 221–263 (2006)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Talagrand, M.: Mean field models for spin glasses. Volume I, vol. 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-Verlag, Berlin, 2011. Basic examplesGoogle Scholar
  46. 46.
    Talagrand, M.: Mean field models for spin glasses. Volume II, vol. 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
  47. 47.
    Toninelli, F.L.: About the Almeida–Thouless transition line in the Sherrington–Kirkpatrick mean-field spin glass model. Europhys. Lett. 60(5), 764–767 (2002)ADSCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of StatisticsStanford UniversityStanfordUSA

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