Phase transition for percolation on a randomly stretched square lattice

Let {& xi;i}i & GE;1 be a sequence of i.i.d. positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in the ith vertical column to another in the (i + 1)th vertical column by an edge having length & xi;i. Then declare independently each edge e in the resulting lattice open with probability pe = p|e| where p & ISIN; [0, 1] and |e| is the length of e. We relate the occurrence of a nontrivial phase transition for this model to moment properties of & xi;1. More precisely, we prove that the model undergoes a nontrivial phase transition when E(& xi;1 & eta;) < & INFIN;, for some & eta; > 1. On the other hand, when E(& xi;1) = & INFIN;, percolation never occurs for p < 1. We also show that the probability of the one-arm event decays no faster than a polynomial in an open interval of parameters p close to the critical point.