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Convergence Time to Equilibrium of the Metropolis Dynamics for the GREM

  • A. M. B. NascimentoEmail author
  • L. R. Fontes
Article
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Abstract

We study the convergence time to equilibrium of the Metropolis dynamics for the generalized random energy model with an arbitrary number of hierarchical levels, a finite and reversible continuous-time Markov process, in terms of the spectral gap of its transition probability matrix. This is done by deducing bounds to the inverse of the gap using a Poincaré inequality and a path technique. We also apply convex analysis tools to give the bounds in the most general case of the model.

Keywords

Spin glasses GREM Metropolis dynamics Convergence to equilibrium Spectral gap Poincaré inequality 

Mathematics Subject Classification

60K35 82B44 82C44 82D30 

Notes

Acknowledgements

This work is part of the Ph.D. thesis of the second author at IME-USP and was supported in part by CNPq 140762/2016-7. We warmfully thank Pierre Picco for suggesting this problem and for innumerable discussions concerning it in many occasions.

References

  1. 1.
    Arous, G.B., Jagannath, A.: Spectral gap estimates in mean field spin glasses. Commun. Math. Phys. 361(1), 1–52 (2018)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bovier, A., Kurkova, I.: Derrida’s generalised random energy models 1: models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Stat. 40(4), 439–480 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezis, H.: Analyse Fonctionnelle. Théorie et Applications. Masson, Fairfax (1983)zbMATHGoogle Scholar
  4. 4.
    Capocaccia, D., Cassandro, M., Picco, P.: On the existence of thermodynamics for the generalized random energy model. J. Stat. Phys. 46(3–4), 493–505 (1987)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cernỳ, J., Wassmer, T.: Aging of the metropolis dynamics on the random energy model. Probab. Theory Relat. Fields 167(1–2), 253–303 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Derrida, B.: A generalization of the random energy model which includes correlations between energies. J. Phys. Lett. 46(9), 401–407 (1985)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of markov chains. Ann. Appl. Probab. 1(1), 36–61 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dorlas, T.C., Dukes, W.M.B.: Large deviation approach to the generalized random energy model. J. Phys. A 35(20), 4385 (2002)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Gayrard, V.: Aging in metropolis dynamics of the REM: a proof. Probab. Theory Relat. Fields 174(1–2), 501–551 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fontes, L.R., Gayrard, V.: Asymptotic behavior and aging of a low temperature cascading 2-GREM dynamics at extreme time scales. (2018) arXiv preprint arXiv:1801.08832
  11. 11.
    Fontes, L.R.G., Isopi, M., Kohayakawa, Y., Picco, P.: The spectral gap of the REM under metropolis dynamics. Ann. Appl. Probab. 8(3), 917–943 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sinclair, A.: Improved Bounds for Mixing Rates of Markov Chains on Combinatorial Structures. Technical Report. Department of Computer Science, University of Edinburgh, Edinburgh (1991)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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