Communications in Mathematical Physics

, Volume 315, Issue 1, pp 241–277 | Cite as

Quenched Limit Theorems for Nearest Neighbour Random Walks in 1D Random Environment

Article
First Online:
Received:
Accepted:

Abstract

It is well known that random walks in a one dimensional random environment can exhibit subdiffusive behavior due to the presence of traps. In this paper we show that the passage times of different traps are asymptotically independent exponential random variables with parameters forming, asymptotically, a Poisson process. This allows us to prove weak quenched limit theorems in the subdiffusive regime where the contribution of traps plays the dominating role.

Keywords

Random Walk Limit Theorem Poisson Process Point Process Central Limit Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aaronson, J.: An introduction to infinite ergodic theory. Math. Surv. & Monographs 50 Providence, RI: Amer. Math. Soc., 1997Google Scholar
  2. 2.
    Bolthausen E., Goldsheid I.: Recurrence and transience of random walks in random environments on a strip. Commun. Math. Phys. 214, 429–447 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Chernov N., Dolgopyat D.: Anomalous current in periodic Lorentz gases with infinite horizon. Russ. Math. Surv. 64, 651–699 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Enriquez, N., Sabot, C., Tournier, L., Zindy, O.: Stable fluctuations for ballistic random walks in random environment on \({\mathbb{Z}}\). http://arXiv.org/abs/preprint1004.1333vL [math.PR], 2010
  5. 5.
    Enriquez, N., Sabot, C., Tournier, L., Zindy, O.: Annealed and quenched fluctuations for ballistic random walks in random environment on Z. PreprintGoogle Scholar
  6. 6.
    Enriquez N., Sabot C., Zindy O.: Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime. Bull. Soc. Math. France 137, 423–452 (2009)MathSciNetMATHGoogle Scholar
  7. 7.
    Enriquez N., Sabot C., Zindy O.: Limit laws for transient random walks in random environment on \({\mathbb{Z} }\). Ann. Institut Fourier 59(6), 2469–2508 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gantert N., Shi Z.: Many visits to a single site by a transient random walk in random environment. Stoch. Proc. Appl. 99, 159–176 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Goldsheid I.: Simple transient random walks in one-dimensional random environment: the central limit theorem. Prob. Th., Rel. Fields 139, 41–64 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Goldsheid I.: Linear and sub-linear growth and the CLT for hitting times of a random walk in random environment on a strip. Prob. Th., Rel. Fields 141, 471–511 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Guivarch, Y.: Heavy tail properties of stationary solutions of multidimensional stochastic recursions. In: Dynamics & stochastics, IMS Lecture Notes Monogr. Ser. 48. Beachwood, OH: Inst. Math. Stat. 2006, pp. 85–99Google Scholar
  12. 12.
    Guivarch Y., Le Page E.: On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks. Erg. Th. Dyn. Sys. 28, 423–446 (2008)MathSciNetGoogle Scholar
  13. 13.
    Kesten H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Kesten H.: Renewal theory for functionals of a Markov chain with general state space. Ann. Prob. 2, 355–386 (1974)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kesten H., Kozlov M., Spitzer F.: A limit law for random walk in a random environment. Compositio Math. 30, 145–168 (1975)MathSciNetMATHGoogle Scholar
  16. 16.
    Peterson, J.: Limiting distributions and large deviations for random walks in random environments. PhD Thesis - University of Minnesota, 2008Google Scholar
  17. 17.
    Peterson, J.: Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment. Ann. Inst. H. Poincaré, Prob. Stat. 45, 685–709 (2009)Google Scholar
  18. 18.
    Peterson, J., Samorodnitsky, G.: Weak quenched limiting distributions for transient one-dimensional random walk in a random environment http://arxiv.org/abs/1011.6366v3 [math.PR], 2010
  19. 19.
    Peterson J., Zeitouni O.: Quenched limits for transient zero-speed one-dimensional random walk in random environment. Ann. Prob. 37, 143–188 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Resnick, S.: Extreme Values, Regular Variation And Point Processes. Belin-Heidelberg-New York: Springer, 2007, 320 ppGoogle Scholar
  21. 21.
    Revesz, P.: Random walk in random and non-random environments. 2nd ed. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2005, xvi+380 ppGoogle Scholar
  22. 22.
    Samorodnitsky, G., Taqqu, M.: Stable non-Gaussian random processes, Boca Ratan, FL: Chapman & Hall/CRC, 1994, 640 ppGoogle Scholar
  23. 23.
    Shi Z.: A local time curiosity in random environment. Stoch. Process. Appl. 76, 231–250 (1998)MATHCrossRefGoogle Scholar
  24. 24.
    Shiga, T., Tanaka, H.: Infinitely divisible random probability distributions with an application to a random motion in a random environment. Electron. J. Probab. 11, paper 44, 1144–1183 (2006)Google Scholar
  25. 25.
    Solomon F.: Random walks in a random environment. Ann. Prob. 3, 1–31 (1975)MATHCrossRefGoogle Scholar
  26. 26.
    Sinai, Ya. G.: The limiting behavior of a one-dimensional random walk in a random medium. Theory Prob. Appl. 27, 256–268 (1982)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Institute of Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.School of Mathematical SciencesQueen Mary University of LondonLondonGreat Britain

Personalised recommendations