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Aging in Metropolis dynamics of the REM: a proof

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Abstract

We consider Metropolis dynamics of the Random Energy Model. We prove that the classical two-time correlation function that allows one to establish aging converges almost surely to the arcsine law distribution function, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. To do this we link the two-time correlation function to a certain continuous-time clock process which, after proper rescaling, is proven to converge to a stable subordinator almost surely in the random environment and in the fine \(J_1\)-topology of Skorohod. This fine topology then enables us to deduce from the arcsine law for stable subordinators the asymptotic behavior of the two-time correlation function that characterizes aging.

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References

  1. Aldous, D.J., Brown, M.: Inequalities for rare events in time-reversible Markov chains. I. In: Shaked, M., Tong, Y.L. (eds.) Stochastic Inequalities (Seattle, WA, 1991). IMS Lecture Notes Monogr. Ser., vol. 22, pp. 1–16. Inst. Math. Statist., Hayward (1992)

    Google Scholar 

  2. Ben Arous, G., Bovier, A., Černý, J., Černý, J.: Universality of the REM for dynamics of mean-field spin glasses. Commun. Math. Phys. 282(3), 663–695 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben Arous, G., Bovier, A., Gayrard, V., Gayrard, V.: Aging in the random energy model. Phys. Rev. Lett. 88(8), 087201 (2002)

    Article  MATH  Google Scholar 

  4. Ben Arous, G., Bovier, A., Gayrard, V., Gayrard, V.: Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Commun. Math. Phys. 235(3), 379–425 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben Arous, G., Bovier, A., Gayrard, V., Gayrard, V.: Glauber dynamics of the random energy model. II. Aging below the critical temperature. Commun. Math. Phys. 236(1), 1–54 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben Arous, G., Černý, J.: The arcsine law as a universal aging scheme for trap models. Commun. Pure Appl. Math. 61(3), 289–329 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bezerra, S.C., Fontes, L.R.G., Gava, R.J., Gayrard, V., Mathieu, P.: Scaling limits and aging for asymmetric trap models on the complete graph and \(K\) processes. ALEA Lat. Am. J Probab. Math. Stat. 9(2), 303–321 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Bouchaud, J.-P., Dean, D.S.: Aging on Parisi’s tree. J. Phys. I (Fr.) 5, 265 (1995)

    Article  Google Scholar 

  10. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Relat. F. 119(1), 99–161 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys. 228(2), 219–255 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bovier, A., Gayrard, V., Švejda, A.: Convergence to extremal processes in random environments and extremal ageing in SK models. Probab. Theory Relat. F. 157(1–2), 251–283 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Černý, J., Wassmer, T.: Aging of the Metropolis dynamics on the random energy model. Probab. Theory Relat. F. 167(1–2), 253–303 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45(2), 79–82 (1980)

    Article  MathSciNet  Google Scholar 

  16. Derrida, B.: A generalization of the random energy model which includes correlations between energies. J. Phys. Lett. 46, 401–407 (1985)

    Article  MathSciNet  Google Scholar 

  17. Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1(1), 36–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Duplantier, B., Halsey, T. C., Rivasseau, V., (eds.) Glasses and Grains, Papers from the 13th Poincaré Seminar held in Paris, November 21, 2009. Progress in Mathematical Physics, vol. 61. Birkhäuser/Springer Basel AG, Basel (2011)

  19. Durrett, R., Resnick, S.I.: Functional limit theorems for dependent variables. Ann. Probab. 6(5), 829–846 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fontes, L.R.G., Mathieu, P.: On the dynamics of trap models in \({\mathbb{Z}}^d\). Proc. Lond. Math. Soc. (3) 108(6), 1562–1592 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gayrard, V.: Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM. Preprint (2010). arXiv:1008.3849

  23. Gayrard, V.: Convergence of clock process in random environments and aging in Bouchaud’s asymmetric trap model on the complete graph. Electron. J. Probab 17(58), 33 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Gayrard, V.: Convergence of clock processes and aging in Metropolis dynamics of a truncated REM. Annales Henri Poincaré 17(3), 537–614 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gayrard, V., Švejda, A.: Convergence of clock processes on infinite graphs and aging in Bouchaud’s asymmetric trap model on \(\mathbb{Z}^d\). ALEA. Lat. Am. J Probab. Math. Stat. 11(2), 781–822 (2015)

    MATH  Google Scholar 

  26. Junier, I., Kurchan, J.: Microscopic realizations of the trap model. J. Phys. A Math. Gen. 37(13), 3945 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Keilson, J.: Markov chain models—rarity and exponentiality. Applied Mathematical Sciences, vol. 28. Springer, New York, Berlin (1979)

    Book  MATH  Google Scholar 

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  1. Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

    Véronique Gayrard

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  1. Véronique Gayrard
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Gayrard, V. Aging in Metropolis dynamics of the REM: a proof. Probab. Theory Relat. Fields 174, 501–551 (2019). https://doi.org/10.1007/s00440-018-0873-6

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  • DOI: https://doi.org/10.1007/s00440-018-0873-6

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