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Random walks in a sparse random environment. (English) Zbl 1354.60121
Summary: 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 [I. S. Sineva, Mosc. Univ. Math. Bull. 39, No. 4, 5–14 (1984); translation from Vestn. Mosk. Univ., Ser. I 1984, No. 4, 5–12 (1984; Zbl 0558.60052)] and [H. E. Stanley and S. Havlin, “Generalisation of the Sinai anomalous diffusion law”, J. Phys. A. 20, No. 9, L615–L618 (1987)].

MSC:
60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
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