Electronic Journal of Probability

Random walks in a sparse random environment

Anastasios Matzavinos, Alexander Roitershtein, and Youngsoo Seol

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We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a “locally strong” perturbation of a simple random walk by a random potential induced by “rare impurities,” which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai’s scaling of $(\log n)^2$ for the location of the random walk after $n$ steps to $(\log n)^\alpha ,$ where $\alpha >0$ is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [29] and [31].

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 72, 20 pp.

Received: 5 September 2016
Accepted: 24 November 2016
First available in Project Euclid: 6 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: primary 60K37 secondary 60F05

RWRE sparse environment limit theorems Sinai’s walk

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Matzavinos, Anastasios; Roitershtein, Alexander; Seol, Youngsoo. Random walks in a sparse random environment. Electron. J. Probab. 21 (2016), paper no. 72, 20 pp. doi:10.1214/16-EJP16. https://projecteuclid.org/euclid.ejp/1480993226

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