A quenched invariance principle for non-elliptic random walk in i.i.d. balanced random environment

Noam Berger, Jean Dominique Deuschel

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

We consider a random walk on ℤd, d ≥ 2, in an i.i.d. balanced random environment, that is a random walk for which the probability to jump from x ∈ ℤd to nearest neighbor x+e is the same as to nearest neighbor x-e. Assuming that the environment is genuinely d-dimensional and balanced we show a quenched invariance principle: for P almost every environment, the diffusively rescaled random walk converges to a Brownian motion with deterministic non-degenerate diffusion matrix. Within the i.i.d. setting, our result extend both Lawler's uniformly elliptic result (Comm Math Phys, 87(1), pp 81-87, 1982/1983) and Guo and Zeitouni's elliptic result (to appear in PTRF, 2010) to the general (non elliptic) case. Our proof is based on analytic methods and percolation arguments.

Original languageEnglish
Pages (from-to)91-126
Number of pages36
JournalProbability Theory and Related Fields
Volume158
Issue number1-2
DOIs
StatePublished - Feb 2014
Externally publishedYes

Keywords

  • Maximum principle
  • Mean value inequality
  • Non-ellipticity
  • Percolation
  • Random walks in random environments

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