On Two-Dimensional Random Walk Among Heavy-Tailed Conductances

Jiří Černý (ETH Zürich)


We consider a random walk among unbounded random conductances on the two-dimensional integer lattice. When the distribution of the conductances has an infinite expectation and a polynomial tail, we show that the scaling limit of this process is the fractional kinetics process. This extends the results of the paper [BC10] where a similar limit statement was proved in dimension larger than two. To make this extension possible, we prove several estimates on the Green function of the process killed on exiting large balls.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 293-313

Publication Date: February 6, 2011

DOI: 10.1214/EJP.v16-849


  1. M. T. Barlow and J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010), 234--276. Math. Review 2599199
  2. M. T. Barlow and B. M. Hambly. Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14 (2009), 1--27. Math. Review 2471657
  3. M. T. Barlow and J. Černý. Convergence to fractional kinetics for random walks associated with unbounded conductances. To appear in Probab. Theory Related Fields, 2011. Math. Review number not available.
  4. M. T. Barlow and X. Zheng. The random conductance model with Cauchy tails. Ann. Appl. Probab. 20 (2010), 869--889. Math. Review 2680551
  5. G. Ben Arous and J. Černý. Bouchaud's model exhibits two aging regimes in dimension one. Ann. Appl. Probab. 15 (2005), 1161--1192. Math. Review 2134101
  6. G. Ben Arous and J. Černý. Dynamics of trap models. Ecole d'Eté de Physique des Houches, Session LXXXIII "Mathematical Statistical Physics", Elsevier, 2006, pp. 331--394. Math. Review 2581889
  7. G. Ben Arous and J. Černý. Scaling limit for trap models on Zd. Ann. Probab. 35 (2007), 2356--2384. Math. Review 2353391
  8. G. Ben Arous, J. Černý and T. Mountford. Aging in two-dimensional Bouchaud's model. Probab. Theory Related Fields 134 (2006), 1--43. Math. Review 2221784
  9. P. Billingsley. Convergence of probability measures. John Wiley & Sons Inc., New York, 1968. Math. Review 0233396
  10. 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--375. Math. Review 2023130
  11. 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. Math. Review 1905852
  12. J.-C. Mourrat. Variance decay for functionals of the environment viewed by the particle. arXiv:0902.0204, to appear in Ann. Inst. H. Poincaré Probab. Statist. Math. Review number not available.
  13. J.-C. Mourrat. Scaling limit of the random walk among random traps on Zd. arXiv:1001.2459, to appear in Ann. Inst. H. Poincaré Probab. Statist. Math. Review number not available.
  14. C. Stone. Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7 (1963), 638--660. Math. Review 0158440

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