A Note about a Burton Keane's Theorem

In BURTON-KEANE'S structural paper 1] a situation has not been lightened, such that the proof of their result about the topological structure of ribbons in the two-dimensional site percolation has a gap. We introduce the connuent zone notion and prove, using Burton-Keane's arguments , the almost surely absence of these zones (This completes the proof of the topological structure of ribbons announced by the authors). In the end we give a alternative proof of the corollary of Theorem 2 in 1]. 1 Deenitions and recalls : We begin by following line by line what BURTON and KEANE have stated in 1]. 1.1 Basic Clustering We will use the classic Euclidean distance d between two points x and y of Z 2 given by the quadratic norm k:k 2. We say that two points of Z 2 are neighbors if they are at distance 1 from each other. We call path a sequence ((nite or not) (x 1 ; :::; x n) such that d(x i ; x i+1) = 1 for all i = 1; :::; n ? 1 and x i 6 = x j if i 6 = j i.e. to us a path is a chain of neighboring points with no loops.