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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