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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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Acknowledgments

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

Author information

Correspondence to Erich Baur.

Additional information

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