On the truncated anisotropic long-range percolation on Z2

Consider the following bond percolation process on Z2: each vertex x∈Z2 is connected to each of its nearest neighbour in the vertical direction with probability pv=ε>0; and in the horizontal direction each vertex x∈Z2 is connected to each of the vertices x±(i,0) with probability pi⩾0, i⩾1, with all different connections being independent. We prove that if piʹs satisfy some regularity property, namely if pi⩾1/i ln i, for i sufficiently large, then for each ε>0 there exists K≡K(ε) such that for truncated percolation process (for which p̃i=pi if i⩽K and p̃j=0 if j>K) the probability of the open cluster of the origin to be infinite remains positive.