On the truncated anisotropic long-range percolation on

Consider the following bond percolation process on : each vertex is connected to each of its nearest neighbour in the vertical direction with probability pv=[var epsilon]>0; and in the horizontal direction each vertex is connected to each of the vertices x±(i,0) with probability pi[greater-or-equal, slanted]0, i[greater-or-equal, slanted]1, with all different connections being independent. We prove that if pi's satisfy some regularity property, namely if pi[greater-or-equal, slanted]1/i ln i, for i sufficiently large, then for each [var epsilon]>0 there exists K[reverse not equivalent]K([var epsilon]) such that for truncated percolation process (for which if i[less-than-or-equals, slant]K and if j>K) the probability of the open cluster of the origin to be infinite remains positive.