Electronic Communications in Probability

Two-site localisation in the Bouchaud trap model with slowly varying traps

Stephen Muirhead

Full-text: Open access


We consider the Bouchaud trap model on the integers in the case that the trap distribution has a slowly varying tail at infinity. We prove that the model eventually localises on exactly two sites with overwhelming probability. This is a stronger form of localisation than has previously been established in the literature for the Bouchaud trap model on the integers in the case of regularly varying traps. Underlying this result is the fact that the sum of a sequence of i.i.d. random variables with a slowly varying tail is asymptotically dominated by the maximal term.<!–EndFragment–>

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 25, 15 pp.

Accepted: 13 March 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60F10: Large deviations 35P05: General topics in linear spectral theory

Bouchaud trap model localisation slowly varying tail

This work is licensed under a Creative Commons Attribution 3.0 License.


Muirhead, Stephen. Two-site localisation in the Bouchaud trap model with slowly varying traps. Electron. Commun. Probab. 20 (2015), paper no. 25, 15 pp. doi:10.1214/ECP.v20-3723. https://projecteuclid.org/euclid.ecp/1465320952

Export citation


  • Ben Arous, G.; Cabezas, M.; ÄŒerný, J.; Royfman, R. Randomly trapped random walks. Ann. Probab. (to appear), 2014.
  • Ben Arous, G.; Fribergh, A. Biased random walks on random graphs. arXiv:1406.5076, 2014.
  • Ben Arous, Gérard; Gün, Onur. Universality and extremal aging for dynamics of spin glasses on subexponential time scales. Comm. Pure Appl. Math. 65 (2012), no. 1, 77–127.
  • Ben Arous, Gérard; ÄŒerný, Jiří. Dynamics of trap models. Math. Stat. Physics Lecture Notes – Les Houches Summer School, 83, 2006.
  • Bertin, E. M.; Bouchaud, J. P. Subdiffusion and localization in the one-dimensional trap model. Phys. Rev., E 67:026128, 2003.
  • Borodin, A. N.; Ibragimov, A. Limit theorems for functionals of random walks. Proceedings of the Steklov Institute of Mathematics, 1994.
  • Bouchaud, J. P. Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France), 2:1705–1713, 1992.
  • Bovier, Anton; Gayrard, Véronique; Å vejda, Adéla. Convergence to extremal processes in random environments and extremal ageing in SK models. Probab. Theory Related Fields 157 (2013), no. 1-2, 251–283.
  • Croydon, D. A.; Fribergh, A.; Kumagai, T. Biased random walk on critical Galton-Watson trees conditioned to survive. Probab. Theory Related Fields 157 (2013), no. 1-2, 453–507.
  • Croydon, David; Muirhead, Stephen. Functional limit theorems for the Bouchaud trap model with slowly varying traps. Stochastic Process. Appl. 125 (2015), no. 5, 1980–2009.
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M. Chaotic time dependence in a disordered spin system. Probab. Theory Related Fields 115 (1999), no. 3, 417–443.
  • Fontes, L. R. G.; Isopi, M.; Newman, C. M. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab. 30 (2002), no. 2, 579–604.
  • Gayrard, Véronique. Convergence of clock process in random environments and aging in Bouchaud's asymmetric trap model on the complete graph. Electron. J. Probab. 17 (2012), no. 58, 33 pp.
  • Gün, O. Extremal aging for trap models. arXiv:1312.1137, 2013.
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1
  • Kasahara, Yuji. A limit theorem for sums of i.i.d. random variables with slowly varying tail probability. J. Math. Kyoto Univ. 26 (1986), no. 3, 437–443.
  • Lamperti, John. On extreme order statistics. Ann. Math. Statist 35 1964 1726–1737.
  • Révész, P. Random Walk in Random and Non-Random Environments. World Scientific, 1983.
  • Whitt, Ward. Stochastic-process limits. An introduction to stochastic-process limits and their application to queues. Springer Series in Operations Research. Springer-Verlag, New York, 2002. xxiv+602 pp. ISBN: 0-387-95358-2