Phase transition for percolation on a randomly stretched square lattice

Abstract

Let { ξ i } i ≥ 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 i -th vertical column to another in the ( i + 1) -th vertical column by an edge having length ξ i . Then declare independently each edge e in the resulting lattice open with probability p e = p | e | where p ∈ [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 ξ 1 . More precisely, we prove that the model undergoes a nontrivial phase transition when E ( ξ η 1 ) < ∞ , for some η > 1 . On the other hand, when E ( ξ 1 ) = ∞ , 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.