Scaling limit of the random walk among random traps on ℤd

Jean-Christophe Mourrat

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 3, page 813-849
  • ISSN: 0246-0203

Abstract

top
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.

How to cite

top

Mourrat, Jean-Christophe. "Scaling limit of the random walk among random traps on ℤd." Annales de l'I.H.P. Probabilités et statistiques 47.3 (2011): 813-849. <http://eudml.org/doc/242850>.

@article{Mourrat2011,
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.},
author = {Mourrat, Jean-Christophe},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk in random environment; trap model; stable process; fractional kinetics},
language = {eng},
number = {3},
pages = {813-849},
publisher = {Gauthier-Villars},
title = {Scaling limit of the random walk among random traps on ℤd},
url = {http://eudml.org/doc/242850},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Mourrat, Jean-Christophe
TI - Scaling limit of the random walk among random traps on ℤd
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 3
SP - 813
EP - 849
AB - 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.
LA - eng
KW - random walk in random environment; trap model; stable process; fractional kinetics
UR - http://eudml.org/doc/242850
ER -

References

top
  1. [1] M. Barlow and J. Černý. Convergence to fractional kinetics for random walks associated with unbounded conductances. Preprint, 2009. Zbl1238.60114MR2776627
  2. [2] M. Barlow and J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010) 234–276. Zbl1189.60187MR2599199
  3. [3] L. E. Baum and M. Katz. Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120 (1965) 108–123. Zbl0142.14802MR198524
  4. [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. Zbl1069.60092MR2134101
  5. [5] G. Ben Arous and J. Černý. Dynamics of trap models. In Les Houches Summer School Lecture Notes 331–394. Elsevier, Amsterdam, 2006. Zbl05723801MR2581889
  6. [6] G. Ben Arous and J. Černý. Scaling limit for trap models on ℤd. Ann. Probab. 35 (2007) 2356–2384. Zbl1134.60064MR2353391
  7. [7] G. Ben Arous, J. Černý and T. Mountford. Aging in two-dimensional Bouchaud’s model. Probab. Theory Related Fields 134 (2006) 1–43. Zbl1089.82017MR2221784
  8. [8] E. Bertin and J.-P. Bouchaud. Subdiffusion and localization in the one-dimensional trap model. Phys. Rev. E 67 (2003) 026128. 
  9. [9] J. Bertoin. Lévy Processes. Cambridge Tracts in Math. 121. Cambridge Univ. Press, 1996. Zbl0861.60003MR1406564
  10. [10] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley, New York, 1968. Zbl0944.60003MR233396
  11. [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. Zbl1079.60079MR2023130
  12. [12] J.-P. Bouchaud. Weak ergodicity breaking and aging in disordered systems. J. Phys. I 2 (1992) 1705–1713. 
  13. [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. [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. Zbl0634.60066MR898496
  15. [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. [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. Zbl0713.60041MR1003538
  17. [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. Zbl1015.60099MR1905852
  18. [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. Zbl1154.60073MR2435851
  19. [19] L. R. G. Fontes, P. Mathieu and M. Vachkovskaia. On the dynamics of trap models in ℤd. To appear. 
  20. [20] J. F. C. Kingman. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 (1968) 499–510. Zbl0182.22802MR254907
  21. [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. [22] R. Lyons, with Y. Peres. Probability on Trees and Networks. Cambridge Univ. Press. To appear. Available at http://mypage.iu.edu/~rdlyons/. 
  23. [23] C. Monthus and J.-P. Bouchaud. Models of traps and glass phenomenology. J. Phys. A Math. Gen. 29 (1996) 3847–3869. Zbl0900.82090
  24. [24] J.-C. Mourrat. Variance decay for functionals of the environment viewed by the particle. Ann. Inst. H. Poincaré Probab. Statist. (2010). To appear. Zbl1213.60163MR2779406
  25. [25] J. Nash. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958) 931–954. Zbl0096.06902MR100158
  26. [26] B. Rinn, P. Maass and J.-P. Bouchaud. Multiple scaling regimes in simple aging models. Phys. Rev. Lett. 84 (2000) 5403–5406. 
  27. [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. [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. [29] W. Whitt. Stochastic-Process Limits. Springer, New York, 2002. Zbl0993.60001MR1876437
  30. [30] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Math. 138. Cambridge Univ. Press, 2000. Zbl0951.60002MR1743100
  31. [31] G. M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002) 461–580. Zbl0999.82053MR1937584

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.