Random walks in a sparse random environment.
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We introduce random walks in a sparse random environment on the integer lattice Z and investigate such fundamental asymptotic property of this model as recurrence-transience criteria, the existence of the asymptotic speed and a phase transition for its value, limit theorems in both transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physics interpretation. More specifically, the random walk in a sparse random environment can be characterized as a perturbation of the simple random walk by a random potential which is induced by &ldquo rare impurities&rdquo randomly distributed over the integer lattice. The &ldquo impurities &rdquo in the media are rigorously defined as a marked point process on Z. The most interesting seems to be the critical (recurrent) case, where Sinai's scaling (log n)^2 for the location of the random walk after n steps is generalized to basically (log n)^&alpha, with &alpha>0 being a parameter determined by the distribution of the distance between two successive impurities of the media.