Electronic Journal of Probability

Random walks with unbounded jumps among random conductances I: Uniform quenched CLT

Christophe Gallesco and Serguei Popov

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Abstract

We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched uniform invariance principle for the random walk. This means that the rescaled trajectory of length n is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval  of length $O(\sqrt{n})$ around the origin.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 85, 22 pp.

Dates
Accepted: 4 October 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062407

Digital Object Identifier
doi:10.1214/EJP.v17-1826

Mathematical Reviews number (MathSciNet)
MR2988400

Zentralblatt MATH identifier
1252.60100

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K37: Processes in random environments

Keywords
ergodic environment unbounded jumps hitting probabilities exit distribution

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Gallesco, Christophe; Popov, Serguei. Random walks with unbounded jumps among random conductances I: Uniform quenched CLT. Electron. J. Probab. 17 (2012), paper no. 85, 22 pp. doi:10.1214/EJP.v17-1826. https://projecteuclid.org/euclid.ejp/1465062407


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References

  • Barlow, Martin T.; Bass, Richard F.; Kumagai, Takashi. Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. 261 (2009), no. 2, 297–320.
  • Barlow, M. T.; Deuschel, J.-D. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010), no. 1, 234–276.
  • Berger, Noam; Biskup, Marek. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007), no. 1-2, 83–120.
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • Biskup, Marek. Recent progress on the random conductance model. Probab. Surv. 8 (2011), 294–373.
  • Biskup, Marek; Prescott, Timothy M. Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 (2007), no. 49, 1323–1348.
  • Caputo, Pietro; Faggionato, Alessandra; Gaudillière, Alexandre. Recurrence and transience for long range reversible random walks on a random point process. Electron. J. Probab. 14 (2009), no. 90, 2580–2616.
  • Comets, Francis; Popov, Serguei. Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Related Fields 126 (2003), no. 4, 571–609.
  • F. Comets, S. Popov (2012) Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards. Ann. Inst. Henri Poincaré Probab. Stat. 48, 721–744.
  • Comets, Francis; Popov, Serguei; Schütz, Gunter M.; Vachkovskaia, Marina. Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. 191 (2009), no. 3, 497–537.
  • Comets, Francis; Popov, Serguei; Schütz, Gunter M.; Vachkovskaia, Marina. Quenched invariance principle for the Knudsen stochastic billiard in a random tube. Ann. Probab. 38 (2010), no. 3, 1019–1061.
  • Comets, Francis; Popov, Serguei; Schütz, Gunter M.; Vachkovskaia, Marina. Knudsen gas in a finite random tube: transport diffusion and first passage properties. J. Stat. Phys. 140 (2010), no. 5, 948–984.
  • Delmotte, Thierry. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999), no. 1, 181–232.
  • C. Gallesco, N. Gantert, S. Popov, M. Vachkovskaia. A conditional quenched CLT for random walks among random conductances on Z^d. arXiv:1108.5616
  • C. Gallesco, S. Popov Random walks with unbounded jumps among random conductances II: Conditional quenched CLT. arXiv:1210.0591
  • Gantert, Nina; Peterson, Jonathon. Maximal displacement for bridges of random walks in a random environment. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 3, 663–678.
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4
  • Mathieu, P. Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 (2008), no. 5, 1025–1046.
  • Mathieu, P.; Piatnitski, A. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2085, 2287–2307.
  • Mörters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8
  • Thorisson, Hermann. Coupling, stationarity, and regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. xiv+517 pp. ISBN: 0-387-98779-7