Random walks in a sparse random environment

Abstract

We introduce random walks in a sparse random environment on 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) α , where α > 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]. MSC2010: primary 60K37; secondary 60F05.

Random walks in a strongly sparse random environment

“…Focussing on the case of strong sparsity we consider a nearest neighbor random walk on the set of integers having jumps ±1 with probability 1/2 at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in for the case of moderate sparsity and in Matzavinos et al (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.2010 Mathematics Subject Classification.…”

“…It is more convenient to discuss limit results for T n rather than X n . Distributional limit theorems for X n and T n are proved in [26] for the case where ξ is P-a.s. bounded (the corresponding environment may be called weakly sparse). Then, as expected, the distribution of ξ does not affect the asymptotic behavior of T n in a significant way.…”

“…Allowance is made for occasional huge gaps between the marked sites which thus form a rather sparse subset of Z. Our main purpose is to reveal new effects generated by this extreme sparsity which are absent in [23] and [6,26]. While the former article treats the nonsparse case in which random drifts are imposed at each site, the latter two papers are concerned with a sparse situation like here, the only difference being that enormous gaps between the marked sites are prohibited.…”

“…Since for the nonmarked sites n (that is, for most of sites) ω n = 1/2, it is natural to call ω a sparse random environment. Following [26] we use the term random walk in sparse random environment (RWSRE) for X as defined above with ω being a sparse random environment. We call the environment ω moderately sparse or strongly sparse depending on whether Eξ < ∞ or…”

“…The behavior of any RWRE is affected by both randomness of the environment and randomness of the walk given the environment. The earlier works [6,23,26] demonstrate that in the nonsparse case randomness of the environment has dominating effect. A remarkable feature of the sparse case is that randomness of the environment and randomness of the walk may contribute to a comparable extent.…”