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Quenched localisation in the Bouchaud trap model with slowly varying traps


We consider the quenched localisation of the Bouchaud trap model on the positive integers in the case that the trap distribution has a slowly varying tail at infinity. Our main result is that for each \(N \in \{2, 3, \ldots \}\) there exists a slowly varying tail such that quenched localisation occurs on exactly N sites. As far as we are aware, this is the first example of a model in which the exact number of localisation sites are able to be ‘tuned’ according to the model parameters. Key intuition for this result is provided by an observation about the sum-max ratio for sequences of independent and identically distributed random variables with a slowly varying distributional tail, which is of independent interest.

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  1. 1.

    Aldous, D.J.: Some inequalities for reversible Markov chains. J. Lond. Math. Soc. (2) 25(3), 564–576 (1982)

  2. 2.

    Ben Arous, G., Gün, O.: Universality and extremal aging for dynamics of spin glasses on subexponential time scales. Commun. Pure Appl. Math. 65, 77–127 (2012)

  3. 3.

    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

  4. 4.

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

  5. 5.

    Croydon, D., Fribergh, A., Kumagai, T.: Biased random walk on critical Galton–Watson trees conditioned to survive. Probab. Theory Related Fields 157, 453–507 (2013)

  6. 6.

    Croydon, D., Muirhead, S.: Functional limit theorems for the Bouchaud trap model with slowly varying traps. Stoch. Process. Appl. 125(5), 1980–2009 (2015)

  7. 7.

    Croydon, D., Muirhead, S.: Quenched localisation in the Bouchaud trap model with regularly varying traps (2016). arXiv:1601.00514

  8. 8.

    Darling, D.A.: The influence of the maximum term in the addition of independent random variables. Trans. Am. Math. Soc. 73, 95–107 (1952)

  9. 9.

    Fontes, L.R.G., Isopi, M., Newman, C.M.: Chaotic time dependence in a disordered spin system. Probab. Theory Related Fields 115(3), 417–443 (1999)

  10. 10.

    Gün, O.: Extremal aging for trap models (2013). arXiv:1312.1137

  11. 11.

    Kasahara, Y.: A limit theorem for sums of i.i.d. random variables with slowly varying tail probability. J. Math. Kyoto Univ. 37, 197–205 (1986)

  12. 12.

    Muirhead, S.: Two-site localisation in the Bouchaud trap model with slowly varying traps. Electron. Commun. Probab. 20(25), 1–15 (2015)

  13. 13.

    Sinaĭ, Y.G.: The limit behavior of a one-dimensional random walk in a random environment. Teor. Veroyatnost. i Primenen. 27, 247–258 (1982)

  14. 14.

    Zeitouni, O.: Random walks in random environment. In: Lectures on probability theory and statistics, Lecture Notes in Mathematics, vol. 1837, pp. 189–312. Springer, Berlin (2004)

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Part of this article was written whilst the first author was a Visiting Associate Professor at Kyoto University, Research Institute for Mathematical Sciences. He would like to thank Takashi Kumagai and Ryoki Fukushima for their kind and generous hospitality. We would like to thank an anonymous referee for their thoughtful comments.

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Correspondence to Stephen Muirhead.

Additional information

S. Muirhead was partially supported by a Graduate Research Scholarship from University College London and the Leverhulme Research Grant RPG-2012-608 held by Nadia Sidorova, and partially supported by the Engineering and Physical Sciences Research Council (EPSRC) Fellowship EP/M002896/1 held by Dmitry Belyaev.

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Croydon, D.A., Muirhead, S. Quenched localisation in the Bouchaud trap model with slowly varying traps. Probab. Theory Relat. Fields 168, 269–315 (2017). https://doi.org/10.1007/s00440-016-0710-8

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  • Random walk in random environment
  • Bouchaud trap model
  • Localisation
  • Slowly varying tail

Mathematics Subject Classification

  • 60K37
  • 82C44
  • 60G50