Electronic Journal of Probability
- Electron. J. Probab.
- Volume 21 (2016), paper no. 72, 20 pp.
Random walks in a sparse random environment
Anastasios Matzavinos, Alexander Roitershtein, and Youngsoo Seol
Abstract
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
Source
Electron. J. Probab., Volume 21 (2016), paper no. 72, 20 pp.
Dates
Received: 5 September 2016
Accepted: 24 November 2016
First available in Project Euclid: 6 December 2016
Permanent link to this document
https://projecteuclid.org/euclid.ejp/1480993226
Digital Object Identifier
doi:10.1214/16-EJP16
Mathematical Reviews number (MathSciNet)
MR3592203
Zentralblatt MATH identifier
1354.60121
Subjects
Primary: primary 60K37 secondary 60F05
Keywords
RWRE sparse environment limit theorems Sinai’s walk
Rights
Creative Commons Attribution 4.0 International License.
Citation
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