On the truncated anisotropic long-range percolation on Z(2)

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.