Phase transition for percolation on a randomly stretched square lattice

Abstract

Let ${\{{\mathit{\xi }}_{\mathit{i}}\}}_{\mathit{i}\ge 1}$ 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 $(\mathit{i}+1)$th vertical column by an edge having length ${\mathit{\xi }}_{\mathit{i}}$. Then declare independently each edge e in the resulting lattice open with probability ${\mathit{p}}_{\mathit{e}}={\mathit{p}}^{|\mathit{e}|}$ where $\mathit{p}\in [0,1]$ and $|\mathit{e}|$ is the length of e. We relate the occurrence of a nontrivial phase transition for this model to moment properties of ${\mathit{\xi }}_1$. More precisely, we prove that the model undergoes a nontrivial phase transition when $\mathbb{E}({\mathit{\xi }}_1^{\mathit{\eta }})<\infty$, for some $\mathit{\eta }>1$. On the other hand, when $\mathbb{E}({\mathit{\xi }}_1)=\infty$, percolation never occurs for $\mathit{p}<1$. 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.