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March 2012 Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment
Alexander Drewitz, Alejandro F. Ramírez
Ann. Probab. 40(2): 459-534 (March 2012). DOI: 10.1214/10-AOP637

Abstract

Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each γ ∈ (0, 1) the ballisticity condition (T)γ and the condition (T') defined as the fulfillment of (T)γ for each γ ∈ (0, 1). Sznitman proved that (T') implies a ballistic law of large numbers. Furthermore, he showed that for all γ ∈ (0.5, 1), (T)γ is equivalent to (T'). Recently, Berger has proved that in dimensions larger than three, for each γ ∈ (0, 1), condition (T)γ implies a ballistic law of large numbers. On the other hand, Drewitz and Ramírez have shown that in dimensions d ≥ 2 there is a constant γd ∈ (0.366, 0.388) such that for each γ ∈ (γd, 1), condition (T)γ is equivalent to (T'). Here, for dimensions larger than three, we extend the previous range of equivalence to all γ ∈ (0, 1). For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.

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Alexander Drewitz. Alejandro F. Ramírez. "Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment." Ann. Probab. 40 (2) 459 - 534, March 2012. https://doi.org/10.1214/10-AOP637

Information

Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1245.60099
MathSciNet: MR2952083
Digital Object Identifier: 10.1214/10-AOP637

Subjects:
Primary: 60K37 , 82D30

Keywords: ballisticity conditions , exit estimates , Random walk in random environment

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 2 • March 2012
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