  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 [28] and [30].
