Advertisement

Journal of Statistical Physics

, Volume 129, Issue 5–6, pp 1081–1116 | Cite as

Replica Symmetry Breaking Condition Exposed by Random Matrix Calculation of Landscape Complexity

  • Yan V. Fyodorov
  • Ian Williams
Article

Abstract

We start with a rather detailed, general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N(≫1)-dimensional Gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity. Then we employ random matrix methods to calculate the density of stationary points, as well as minima, of the associated energy surface. This is used to show that for a generic smooth, concave confining potentials the condition of the zero-temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape, and the mean number of minima are exponential in N. For such systems the (annealed) complexity of minima vanishes cubically when approaching the critical confinement, whereas the cumulative annealed complexity vanishes quadratically. Different behaviour reported in our earlier short communication (Fyodorov et al. in JETP Lett. 85:261, 2007) was due to non-analyticity of the hard-wall confinement potential. Finally, for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points. In particular, we show that in the vicinity of critical confinement the saddle-points with a positive annealed complexity must be close to minima, as they must have a vanishing fraction of negative eigenvalues in the Hessian.

Keywords

Stationary Point Stat Phys Energy Landscape Random Potential Eigenvalue Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mezard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987) MATHGoogle Scholar
  2. 2.
    Parisi, G.: E-preprint arXiv: 0706.0094 [cond-mat.dis-nn] (2007, to appear), in Les Houches Summer School Complex Systems, Eslevier Google Scholar
  3. 3.
    de Almeida, J.R.L., Thouless, D.J.: J. Phys. A 11, 983 (1978) CrossRefADSGoogle Scholar
  4. 4.
    Thouless, D.J., Anderson, P.W., Palmer, R.G.: Philos. Mag. 35, 593 (1977) CrossRefADSGoogle Scholar
  5. 5.
    Talagrand, M.: C. R. Acad. Sci. Ser. I: Math. 337, 111 (2003), and Ann. Math. 163, 221 (2006) MATHMathSciNetGoogle Scholar
  6. 6.
    Talagrand, M.: Probab. Theory Relat. Fields 134, 339 (2006) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Guerra, F.: Commun. Math. Phys. 233, 1 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Aizenman, M., Sims, R., Starr, S.L.: Mean-field spin glass models from the cavity-ROST perspective. E-preprint arXiv: math-ph/0607060 (2006) Google Scholar
  9. 9.
    Mezard, M., Parisi, G.: J. Phys. A: Math. Gen. 23, L1229 (1990) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Mezard, M., Parisi, G.: J. Phys. I France 1, 809 (1991) CrossRefGoogle Scholar
  11. 11.
    Mezard, M., Parisi, G.: J. Phys. I France 2, 2231 (1992) CrossRefGoogle Scholar
  12. 12.
    Engel, A.: Nucl. Phys. B 410, 617 (1993) CrossRefADSGoogle Scholar
  13. 13.
    Franz, S., Mezard, M.: Physica A 210, 48 (1994) CrossRefADSGoogle Scholar
  14. 14.
    Cugliandolo, L.F., Le Doussal, P.: Phys. Rev. E 53, 1525 (1996) CrossRefADSGoogle Scholar
  15. 15.
    Fyodorov, Y.V., Sommers, H.-J.: Nucl. Phys. B [FS] 764, 128 (2007), e-preprint arXiv: cond-mat/0610035 (2006) MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Fyodorov, Y.V., Bouchaud, J.P.: On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces. E-preprint arXiv: 0706.3776 [cond-mat.dis-nn] (2007) Google Scholar
  17. 17.
    Derrida, B.: Phys. Rev. B 24, 2613 (1981) CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Derrida, B.: J. Phys. Lett. 46, 401 (1985) CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Derrida, B., Gardner, E.: J. Phys. C 19, 2253 (1986), and 19, 5783 (1986) CrossRefADSGoogle Scholar
  20. 20.
    Derrida, B., Spohn, H.: J. Stat. Phys. 51, 817 (1988) MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Carpentier, D., Le Doussal, P.: Phys. Rev. E 63, 026110 (2001) CrossRefADSGoogle Scholar
  22. 22.
    Balents, L., Bouchaud, J.P., Mezard, M.: J. Phys. I (France) 6, 1007 (1996) CrossRefGoogle Scholar
  23. 23.
    Touya, C., Dean, D.S.: J. Phys. A 40, 919 (2007) MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Parisi, G.: In: Bovier, A., et al. (eds.) Lecture Notes of the Les Houches Summer School. Elsevier, Amsterdam (2006). E-preprint arXiv: cond-mat/0602349 Google Scholar
  25. 25.
    Aspelmeier, T., Bray, A.J., Moore, M.A.: Phys. Rev. Lett. 92, 087203 (2004) CrossRefADSGoogle Scholar
  26. 26.
    Crisanti, A., Leuzzi, L., Parisi, G., Rizzo, T.: Phys. Rev. B 70, 064423 (2004) CrossRefADSGoogle Scholar
  27. 27.
    Parisi, G., Rizzo, T.: J. Phys. A 37, 7979 (2004) MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Cavagna, A., Giardina, I., Parisi, G.: Phys. Rev. Lett. 92, 120603 (2004) CrossRefADSGoogle Scholar
  29. 29.
    Aspelmeier, T., Blythe, R.A., Bray, A.J., Moore, M.A.: Phys. Rev. B 74, 184411 (2006) CrossRefADSGoogle Scholar
  30. 30.
    Müller, M., Leuzzi, L., Crisanti, A.: Phys. Rev. B 74, 134431 (2006) CrossRefADSGoogle Scholar
  31. 31.
    Engel, A.: J. Phys. Lett. 46, L409 (1985) CrossRefGoogle Scholar
  32. 32.
    Villain, J.: J. Phys. A: Math. Gen. 21, L1099 (1988) CrossRefADSGoogle Scholar
  33. 33.
    Nattermann, T.: In: Young, A.P. (ed.) Spin Glasses and Random Fields, p. 277. World Scientific, Singapore (1998) Google Scholar
  34. 34.
    Le Doussal, P., Monthus, C.: Physica A 317, 143 (2003) CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Cavagna, A., Garrahan, J.P., Giardina, I.: Phys. Rev. E 59, 2808 (1999) CrossRefADSGoogle Scholar
  36. 36.
    Kac, M.: Bull. Am. Math. Soc. 49, 314 (1943) MATHCrossRefGoogle Scholar
  37. 37.
    Rice, S.O.: Mathematical analysis of random noise. In: Selected Papers on Noise and Stochastic Processes. Dover, New York (1954) Google Scholar
  38. 38.
    Belyaev, Ju.K.: Sov. Math. Dokl. 8, 1107 (1967) MATHGoogle Scholar
  39. 39.
    Cline, J.M., Politzer, H.D.: Rey, S.-Y., Wise, M.B.: Commun. Math. Phys. 112, 217 (1987) MATHCrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Adler, R.J., Taylor, J.: Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York (2007) MATHGoogle Scholar
  41. 41.
    Kurchan, J.: J. Phys. A 24, 4969 (1991) CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Longuet-Higgins, M.S.: J. Opt. Soc. Am. 50, 845 (1957) ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Weinrib, A., Halperin, B.I.: Phys. Rev. B 26, 1362 (1982) CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Halperin, B.I., Lax, M.: Phys. Rev. 148, 722 (1966) CrossRefADSGoogle Scholar
  45. 45.
    Vogel, H., Möhring, W.: Density of critical points for a Gaussian random function. E-preprint arXiv: 0707.0457 [physics.flu-dyn] (2007) Google Scholar
  46. 46.
    Broderix, K., Bhattacharya, K.K., Cavagna, A., Zippelius, A., Giardina, I.: Phys. Rev. Lett. 85, 5360 (2000) CrossRefADSGoogle Scholar
  47. 47.
    Doye, J.P.K., Wales, D.J.: J. Chem. Phys. 116, 3777 (2002) CrossRefADSGoogle Scholar
  48. 48.
    Grigera, T.S., Cavagna, A., Giardina, I., Parisi, G.: Phys. Rev. Lett. 88, 055502 (2002) CrossRefADSGoogle Scholar
  49. 49.
    Grigera, T.S.: J. Chem. Phys. 124, 064502 (2006) CrossRefADSGoogle Scholar
  50. 50.
    Douglas, M.R., Shiffman, B., Zelditch, S.: Commun. Math. Phys. 252, 325 (2004), and ibid 265, 617 (2006) MATHCrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Fyodorov, Y.V.: Phys. Rev. Lett. 92, 240601 (2004). Erratum: ibid. 93, 149901 (2004) and Acta Phys. Pol. B 36, 2699 (2005) CrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Mehta, M.L.: Random Matrices. 3rd edn. Elsevier, Amsterdam (2004) MATHGoogle Scholar
  53. 53.
    Azaïs, J.-M., Wschebor, M.: A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. E-preprint arXiv: math.PR/0607041 (2006) Google Scholar
  54. 54.
    Bray, A.J., Dean, D.S.: Phys. Rev. Lett. 98, 150201 (2007) CrossRefADSGoogle Scholar
  55. 55.
    Fyodorov, Y.V., Sommers, H.-J., Williams, I.: JETP Letters 85, 261 (2007) CrossRefADSGoogle Scholar
  56. 56.
    Ben Arous, G., Dembo, A., Guionnet, A.: Probab. Theory Relat. Fields 136, 619 (2006) MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Castellani, T., Cavagna, A.: J. Stat. Mech: Theor. Exp. P05012 (2005) Google Scholar
  58. 58.
    Boutet de Monvel, A., Pastur, L., Shcherbina, M.: J. Stat. Phys. 79, 585 (1995) MATHCrossRefMathSciNetADSGoogle Scholar
  59. 59.
    Dean, D.S., Majumdar, S.N.: Phys. Rev. Lett. 97, 160201 (2006) CrossRefADSMathSciNetGoogle Scholar
  60. 60.
    Ergün, G., Fyodorov, Y.V.: Phys.Rev. E 68, 046124 (2003) CrossRefADSMathSciNetGoogle Scholar
  61. 61.
    Muskhelishvili, N.I.: Singular Integral Equations. Dover, New York (1992) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamEngland

Personalised recommendations