Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Scaling limit of the random walk among random traps on ℤd

Jean-Christophe Mourrat

Full-text: Open access

Abstract

Attributing a positive value τx to each x∈ℤd, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τx), often known as “Bouchaud’s trap model.” We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d≥5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as the time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.

Résumé

Après avoir attribué une valeur positive τx à chaque x de ℤd, nous nous intéressons à une marche aléatoire au plus proche voisin et réversible pour la mesure de poids (τx), souvent appelée ≪ modèle de Bouchaud ≫. Nous supposons que ces poids sont des variables aléatoires indépendantes, de même loi non-intégrable (à queue polynomiale), et que d≥5. Nous identifions, pour presque toute réalisation des (τx), la limite sous-diffusive de ce modèle. Nous commençons la preuve en exprimant la marche aléatoire comme le changement de temps d’une marche aléatoire en conductances aléatoires. Nous nous consacrons ensuite à montrer que ce changement de temps converge, sous la loi moyennée, vers un subordinateur stable. Nous y parvenons en utilisant un résultat antérieur concernant les propriétés de mélange de l’environnement vu par la marche changée de temps.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 3 (2011), 813-849.

Dates
First available in Project Euclid: 23 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1308834860

Digital Object Identifier
doi:10.1214/10-AIHP387

Mathematical Reviews number (MathSciNet)
MR2841076

Zentralblatt MATH identifier
1262.60098

Subjects
Primary: 60K37: Processes in random environments 60G52: Stable processes 60F17: Functional limit theorems; invariance principles 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walk in random environment Trap model Stable process Fractional kinetics

Citation

Mourrat, Jean-Christophe. Scaling limit of the random walk among random traps on ℤ d. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 3, 813--849. doi:10.1214/10-AIHP387. https://projecteuclid.org/euclid.aihp/1308834860


Export citation

References

  • [1] M. Barlow and J. Černý. Convergence to fractional kinetics for random walks associated with unbounded conductances. Preprint, 2009.
  • [2] M. Barlow and J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010) 234–276.
  • [3] L. E. Baum and M. Katz. Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 (1965) 108–123.
  • [4] G. Ben Arous and J. Černý. Bouchaud’s model exhibits two different aging regimes in dimension one. Ann. Appl. Probab. 15 (2005) 1161–1192.
  • [5] G. Ben Arous and J. Černý. Dynamics of trap models. In Les Houches Summer School Lecture Notes 331–394. Elsevier, Amsterdam, 2006.
  • [6] G. Ben Arous and J. Černý. Scaling limit for trap models on ℤd. Ann. Probab. 35 (2007) 2356–2384.
  • [7] G. Ben Arous, J. Černý and T. Mountford. Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134 (2006) 1–43.
  • [8] E. Bertin and J.-P. Bouchaud. Subdiffusion and localization in the one-dimensional trap model. Phys. Rev. E 67 (2003) 026128.
  • [9] J. Bertoin. Lévy Processes. Cambridge Tracts in Math. 121. Cambridge Univ. Press, 1996.
  • [10] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1968.
  • [11] E. Bolthausen and A.-S. Sznitman. On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9 (2002) 345–376.
  • [12] J.-P. Bouchaud. Weak ergodicity breaking and aging in disordered systems. J. Phys. I 2 (1992) 1705–1713.
  • [13] J.-P. Bouchaud, L. Cugliandolo, J. Kurchan and M. Mézard. Out of equilibrium dynamics in spin-glasses and other glassy systems. In Spin Glasses and Random Fields. A. P. Young (Ed.). Series on Directions in Condensed Matter Physics 12. World Scientific, Singapore, 1997.
  • [14] E. A. Carlen, S. Kusuoka and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 245–287.
  • [15] C. De Dominicis, H. Orland and F. Lainée. Stretched exponential relaxation in systems with random free energies. J. Phys.Lett. 46 (1985) L463–L466.
  • [16] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989) 787–855.
  • [17] L. R. G. Fontes, M. Isopi and C. M. Newman. Random walks with strongly inhomogeneous rates and singular diffusions: Convergence, localization and aging in one dimension. Ann. Probab. 30 (2002) 579–604.
  • [18] L. R. G. Fontes and P. Mathieu. K-processes, scaling limit and aging for the trap model in the complete graph. Ann. Probab. 36 (2008) 1322–1358.
  • [19] L. R. G. Fontes, P. Mathieu and M. Vachkovskaia. On the dynamics of trap models in ℤd. To appear.
  • [20] J. F. C. Kingman. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 (1968) 499–510.
  • [21] L. Lundgren, P. Svedlindh, P. Nordblad and O. Beckman. Dynamics of the relaxation-time spectrum in a CuMn spin-glass. Phys. Rev. Lett. 51 (1983) 911–914.
  • [22] R. Lyons, with Y. Peres. Probability on Trees and Networks. Cambridge Univ. Press. To appear. Available at http://mypage.iu.edu/~rdlyons/.
  • [23] C. Monthus and J.-P. Bouchaud. Models of traps and glass phenomenology. J. Phys. A Math. Gen. 29 (1996) 3847–3869.
  • [24] J.-C. Mourrat. Variance decay for functionals of the environment viewed by the particle. Ann. Inst. H. Poincaré Probab. Statist. (2010). To appear.
  • [25] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958) 931–954.
  • [26] B. Rinn, P. Maass and J.-P. Bouchaud. Multiple scaling regimes in simple aging models. Phys. Rev. Lett. 84 (2000) 5403–5406.
  • [27] B. Rinn, P. Maass and J.-P. Bouchaud. Hopping in the glass configuration space: Subaging and generalized scaling laws. Phys. Rev. B 64 (2001) 104417.
  • [28] E. Vincent, J. Hammann, M. Ocio, J.-P. Bouchaud and L. Cugliandolo. Slow Dynamics and Aging in Spin Glasses 184–219. Lecture Notes in Phys. 492. Springer, Berlin, 1997.
  • [29] W. Whitt. Stochastic-Process Limits. Springer, New York, 2002.
  • [30] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Math. 138. Cambridge Univ. Press, 2000.
  • [31] G. M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002) 461–580.