On a Many-Dimensional Random Walk in a Rarefied Random Environment

#### M.V. Menshikov, S.Yu. Popov, V. Sisko, M. Vachkovskaia

2004, v.10, №1, 137-160

ABSTRACT

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 constructing a random environment.
Then, based on the asymptotic behaviour of the function $p(x)$,
we give some sufficient conditions for transience and recurrence.

Keywords: recurrence, transience, Lyapunov functions, random environment