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January 2009 Quenched limits for transient, zero speed one-dimensional random walk in random environment
Jonathon Peterson, Ofer Zeitouni
Ann. Probab. 37(1): 143-188 (January 2009). DOI: 10.1214/08-AOP399

Abstract

We consider a nearest-neighbor, one dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that Xn is of order ns for some s<1. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences {nk} and {xk} depending on the environment only, such that Xnkxk=o(log nk)2 (a localized regime). On the other hand, there exist sequences {tm} and {sm} depending on the environment only, such that logsm/log tms<1 and Pω(Xtm/smx)→1/2 for all x>0 and →0 for x≤0 (a spread out regime).

Citation

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Jonathon Peterson. Ofer Zeitouni. "Quenched limits for transient, zero speed one-dimensional random walk in random environment." Ann. Probab. 37 (1) 143 - 188, January 2009. https://doi.org/10.1214/08-AOP399

Information

Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1179.60070
MathSciNet: MR2489162
Digital Object Identifier: 10.1214/08-AOP399

Subjects:
Primary: 60K37
Secondary: 60F05, 82C41, 82D30

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 1 • January 2009
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