Phase transition for percolation on a randomly stretched lattice

Abstract: Let ${\xi_i}_{i \geq 1}$ 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 $\xi_i$. Then declare independently each edge $e$ in the resulting lattice open with probability $p_e=p<sup>{|e|}$ where $p\in[0,1]$ and $|e|$ is the length of $e$. We relate the occurrence of 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 $\mathbb{E}(\xi_1<sup>\eta)&lt;\infty$, for some $\eta&gt;1$ whereas, when $\mathbb{E}(\xi_1<sup>\eta)=\infty$ for some $\eta&lt;1$, no phase transition occurs.