We consider a modification of the Simple Random Walk (SRW), which can be
described as follows. Initially, any $x\in\Z^d$ becomes "special"
with probability $p(x)$; then, in all special sites
we modify the transition probabilities in order to create
a drift which is directed outwards the origin (in the case of one-
or two-dimensional SRW) or towards the origin
(for higher dimensions), thus con
structing
a random environment.
Then, based on the asymptotic behaviour of the function $p(x)$,
we give some sufficient conditions for transience and recurrence.

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