Abstract
We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove that the scaling limit of this process is a Fractional-Kinetics process—that is the time change of a d-dimensional Brownian motion by the inverse of an independent α-stable subordinator. We further show that the same process appears in the scaling limit of the non-symmetric Bouchaud’s trap model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ben Arous G., Černý J.: Scaling limit for trap models on \({{\mathbb Z^d}}\) . Ann. Probab. 35(6), 2356–2384 (2007)
Zaslavsky G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)
Zaslavsky G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)
Gorenflo R., Mainardi F.: Fractional diffusion processes: probability distributions and continuous time random walk. In: Rangarajan, G., Ding, M. (eds) Processes with Long Range Correlations, Lecture Notes in Physics, no. 621, pp. 148–166. Springer, Berlin (2003)
Hilfer, R. (eds): Applications of Fractional Calculus in Physics. World Scientific Publishing Co. Inc., River Edge (2000)
Metzler R., Klafter J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 77 (2000)
Montroll, E.W., Shlesinger, M.F.: On the wonderful world of random walks. In: Nonequilibrium Phenomena, II. Stud. Stat. Mech. XI, pp. 1–121. North-Holland, Amsterdam (1984)
Shlesinger M.F., Zaslavsky G.M., Klafter J.: Strange kinetics. Nature 363, 31–37 (1993)
Kipnis C., Varadhan S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104(1), 1–19 (1986)
Kozlov S.M.: The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40(2(242)), 61–120, 238 (1985)
De Masi A., Ferrari P.A., Goldstein S., Wick W.D.: An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Stat. Phys. 55(3–4), 787–855 (1989)
Sidoravicius V., Sznitman A.-S.: Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields 129(2), 219–244 (2004)
Biskup M., Prescott T.M.: Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12(49), 1323–1348 (2007) (electronic)
Mathieu P.: Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130(5), 1025–1046 (2008)
Barlow M.T.: Random walks on supercritical percolation clusters. Ann. Probab. 32(4), 3024–3084 (2004)
Berger N., Biskup M.: Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137(1–2), 83–120 (2007)
Mathieu P., Piatnitski A.: Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2085), 2287–2307 (2007)
Barlow, M.T., Deuschel, J.-D.: Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. (2009, to appear)
Bouchaud J.-P.: Weak ergodicity breaking and aging in disordered systems. J. Phys. I (France) 2, 1705–1713 (1992)
Bouchaud J.-P., Dean D.S.: Aging on Parisi’s tree. J. Phys. I (France) 5, 265 (1995)
Rinn B., Maass P., Bouchaud J.-P.: Hopping in the glass configuration space: subaging and generalized scaling laws. Phys. Rev. B 64, 104417 (2001)
Ben Arous G., Černý J.: Bouchaud’s model exhibits two aging regimes in dimension one. Ann. Appl. Probab. 15(2), 1161–1192 (2005)
Ben Arous G., Bovier A., Gayrard V.: Glauber dynamics of the random energy model. I. Metastable motion on the extreme states. Commun. Math. Phys. 235(3), 379–425 (2003)
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(2), 579–604 (2002)
Ben Arous G., Černý J., Mountford T.: Aging in two-dimensional Bouchaud’s model. Probab. Theory Relat. Fields 134(1), 1–43 (2006)
Černý, J.: On Two Properties of Strongly Disordered Systems, Aging and Critical Path Analysis. Ph.D. thesis, EPF Lausanne (2003)
Ben Arous G., Černý J.: The arcsine law as a universal aging scheme for trap models. Commun. Pure Appl. Math. 61(3), 289–329 (2008)
Barlow, M.T., Zheng, X.: The random conductance model with Cauchy tails. Ann. Appl. Probab. (2009, to appear)
Barlow, M.T.: Aspects of first passage percolation. In: École d’été de Probabilités de Saint-Flour, XXV—1995. Lecture Notes in Mathematics, vol. 1690, pp. 1–121. Springer, Berlin (1998)
Kigami K.: Diffusions on Fractals. Cambridge University Press, Cambridge (2001)
Saichev A.I., Zaslavsky G.M.: Fractional kinetic equations: solutions and applications. Chaos 7(4), 753–764 (1997)
Meerschaert M.M., Scheffler H.-P.: Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41(3), 623–638 (2004)
Barlow M.T., Hambly B.M.: Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14, 1–27 (2009)
Whitt, W.: Stochastic-process limits: an introduction to stochastic-process limits and their application to queues. In: Springer Series in Operations Research. Springer, New York (2002)
Kesten, H.: Aspects of first passage percolation. In: École d’été de Probabilités de Saint-Flour, XIV—1984, Lecture Notes in Mathematics, vol. 1180, pp. 125–264. Springer, Berlin (1986)
Liggett T.M., Schonmann R.H., Stacey A.M.: Domination by product measures. Ann. Probab. 25(1), 71–95 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by NSERC (Canada).
An erratum to this article can be found at http://dx.doi.org/10.1007/s00440-011-0344-9
Rights and permissions
About this article
Cite this article
Barlow, M.T., Černý, J. Convergence to fractional kinetics for random walks associated with unbounded conductances. Probab. Theory Relat. Fields 149, 639–673 (2011). https://doi.org/10.1007/s00440-009-0257-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0257-z