Scaling limit of the random walk among random traps on ${\mathbb{Z}}^d$

Mourrat, Jean-Christophe

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 813-849.

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.

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.

[1] M. Barlow and J. Černý. Convergence to fractional kinetics for random walks associated with unbounded conductances. Preprint, 2009. | MR

[2] M. Barlow and J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010) 234-276. | MR | Zbl

[3] L. E. Baum and M. Katz. Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 (1965) 108-123. | MR | Zbl

[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. | MR | Zbl

[5] G. Ben Arous and J. Černý. Dynamics of trap models. In Les Houches Summer School Lecture Notes 331-394. Elsevier, Amsterdam, 2006. | MR

[6] G. Ben Arous and J. Černý. Scaling limit for trap models on ℤd. Ann. Probab. 35 (2007) 2356-2384. | MR | Zbl

[7] G. Ben Arous, J. Černý and T. Mountford. Aging in two-dimensional Bouchaud's model. Probab. Theory Related Fields 134 (2006) 1-43. | MR | Zbl

[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. | MR | Zbl

[10] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1968. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[24] J.-C. Mourrat. Variance decay for functionals of the environment viewed by the particle. Ann. Inst. H. Poincaré Probab. Statist. (2010). To appear. | Numdam | MR | Zbl

[25] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958) 931-954. | MR | Zbl

[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. | MR | Zbl

[30] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Math. 138. Cambridge Univ. Press, 2000. | MR | Zbl

[31] G. M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002) 461-580. | MR | Zbl