The Annals of Probability

Quenched limits for transient, zero speed one-dimensional random walk in random environment

Jonathon Peterson and Ofer Zeitouni

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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).

Article information

Source
Ann. Probab., Volume 37, Number 1 (2009), 143-188.

Dates
First available in Project Euclid: 17 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1234881687

Digital Object Identifier
doi:10.1214/08-AOP399

Mathematical Reviews number (MathSciNet)
MR2489162

Zentralblatt MATH identifier
1179.60070

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F05: Central limit and other weak theorems 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walk random environment

Citation

Peterson, Jonathon; Zeitouni, Ofer. Quenched limits for transient, zero speed one-dimensional random walk in random environment. Ann. Probab. 37 (2009), no. 1, 143--188. doi:10.1214/08-AOP399. https://projecteuclid.org/euclid.aop/1234881687


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