An invariance principle for a class of non-ballistic random walks in random environment

Abstract

We are concerned with random walks on \({\mathbb {Z}}^d\), \(d\ge 3\), in an i.i.d. random environment with transition probabilities \(\varepsilon \)-close to those of simple random walk. We assume that the environment is balanced in one fixed coordinate direction, and invariant under reflection in the coordinate hyperplanes. The invariance condition was used in Baur and Bolthausen (Ann Probab 2013, arXiv:1309.3169) as a weaker replacement of isotropy to study exit distributions. We obtain precise results on mean sojourn times in large balls and prove a quenched invariance principle, showing that for almost all environments, the random walk converges under diffusive rescaling to a Brownian motion with a deterministic (diagonal) diffusion matrix. Our work extends the results of Lawler (Commun Math Phys 87:81–87, 1982), where it is assumed that the environment is balanced in all coordinate directions.

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References

  1. 1.

    Baur, E.: Long-time behavior of random walks in random environment. Part of the Ph.D. thesis, University of Zurich. arXiv:1309.3419 (2013)

  2. 2.

    Baur, E., Bolthausen, E.: Exit laws from large balls of (an)isotropic random walks in random environment. Ann. Probab. arXiv:1309.3169 (2013) (to appear, preprint)

  3. 3.

    Berger, N.: Slowdown estimates for ballistic random walk in random environment. J. Eur. Math. Soc. 14(1), 127-174 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Berger, N., Deuschel, J.-D.: A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment. Probab. Theory Relat. Fields 158, 91-126 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Berger, N., Drewitz, A., Ramírez, A.F.: Effective polynomial ballisticity conditions for random walk in random environment. Commun. Pure Appl. Math. 76(12), 1947-1973 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    Google Scholar 

  7. 7.

    Bolthausen, E., Sznitman, A.-S.: On the static and dynamic points of view for certain random walks in random environment. Methods Appl. Anal. 9(3), 345-376 (2002)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bolthausen, E., Sznitman, A.-S., Zeitouni, O.: Cut points and diffusive random walks in random environment. Ann. Inst. H. Poincaré Probab. Statist. 39, 527-555 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Bolthausen, E., Zeitouni, O.: Multiscale analysis of exit distributions for random walks in random environments. Probab. Theory Relat. Fields 138, 581-645 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bricmont, J., Kupiainen, A.: Random walks in asymmetric random environments. Commun. Math. Phys. 142, 345-420 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Deuschel, J.-D., Guo, X., Ramírez, A. F.: Quenched invariance principle for random walk in time-dependent balanced random environment. arXiv:1503.01964 (2015) (preprint)

  12. 12.

    Drewitz, A., Ramírez, A.F.: Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment. Ann. Probab. 40(2), 459-534 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Drewitz, A., Ramírez, A. F.: Selected topics in random walks in random environments. Topics in percolative and disordered systems. In: Springer Proceedings in Mathematics and Statistics, vol. 69, pp. 23-83. Springer, New York (2014)

  14. 14.

    Guo, X., Zeitouni, O.: Quenched invariance principle for random walks in balanced random environment. Probab. Theory Relat. Fields 152, 207-230 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Kuo, H.J., Trudinger, N.S.: Linear elliptic difference inequalities with random coefficients. Math. Comput. 55, 37-53 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lawler, G.F.: Weak convergence of a random walk in a random environment. Commun. Math. Phys. 87, 81-87 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  18. 18.

    Sznitman, A.-S.: On a class of transient random walks in random environment. Ann. Probab. 29(2), 724-765 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Sznitman, A.-S.: An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Relat. Fields 122(4), 509-544 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Sznitman, A.-S.: On new examples of ballistic random walks in random environment. Ann. Probab. 31(1), 285-322 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Sznitman, A.-S.: Topics in random walks in random environment. In: School and Conference on Probability Theory, ICTP Lecture Notes Series, Trieste, vol. 17, pp. 203-266 (2004)

  22. 22.

    Sznitman, A.-S., Zeitouni, O.: An invariance principle for isotropic diffusions in random environment. Invent. Math. 164, 455-567 (2006)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

I would like to thank Erwin Bolthausen, Jean-Christophe Mourrat and Ofer Zeitouni for helpful discussions, and two anonymous referees for valuable comments.

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Correspondence to Erich Baur.

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Acknowledgment of support. This research was supported by the Swiss National Science Foundation Grant P2ZHP2_151640.

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Baur, E. An invariance principle for a class of non-ballistic random walks in random environment. Probab. Theory Relat. Fields 166, 463–514 (2016). https://doi.org/10.1007/s00440-015-0664-2

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Keywords

  • Random walk in random environment
  • Invariance principle
  • Non-ballistic behavior
  • Perturbative regime
  • Balanced

Mathematics Subject Classification

  • Primary 60K37
  • Secondary 82C41