Consider the following bond percolation process on Z(2): each vertex x is a
n element of Z(2) is connected to each of its nearest neighbour in the vert
ical direction with probability p(upsilon) = epsilon > 0; and in the horizo
ntal direction each vertex x is an element of Z(2) is connected to each of
the vertices x +/- (i, 0) with probability p(i) greater than or equal to 0,
i greater than or equal to 1, with all different connections being indepen
dent. We prove that if p(i)'s satisfy some regularity property, namely if p
(i) greater than or equal to 1/i ln i, for i sufficiently large, then for e
ach epsilon > 0 there exists K = K(epsilon) such that for truncated percola
tion process (for which (p) over tilde(i) = p(i) if i less than or equal to
K and (p) over tilde(j) = 0 if j > K) the probability of the open cluster
of the origin to be infinite remains positive. (C) 1999 Elsevier Science B.
V. All rights reserved.