On the truncated anisotropic long-range percolation on Z2

Abstract

Consider the following bond percolation process on Z 2 : each vertex x∈ Z 2 is connected to each of its nearest neighbour in the vertical direction with probability p v = ε >0; and in the horizontal direction each vertex x∈ Z 2 is connected to each of the vertices x ±( i ,0) with probability p i ⩾0, i⩾1 , with all different connections being independent. We prove that if p i 's satisfy some regularity property, namely if p i ⩾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 p ̃ i =p i if i ⩽ K and p ̃ j =0 if j > K ) the probability of the open cluster of the origin to be infinite remains positive.