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2016 Random walks in a sparse random environment
Anastasios Matzavinos, Alexander Roitershtein, Youngsoo Seol
Electron. J. Probab. 21: 1-20 (2016). DOI: 10.1214/16-EJP16


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].


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Anastasios Matzavinos. Alexander Roitershtein. Youngsoo Seol. "Random walks in a sparse random environment." Electron. J. Probab. 21 1 - 20, 2016.


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

zbMATH: 1354.60121
MathSciNet: MR3592203
Digital Object Identifier: 10.1214/16-EJP16

Primary: Primary 60K37 , secondary 60F05

Keywords: limit theorems , RWRE , Sinai’s walk , sparse environment


Vol.21 • 2016
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