Phase transition for percolation on a randomly stretched lattice

Abstract

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 $i$-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 $p_e=p^{|e|}$ 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 $\mathbb{E}({\xi }_1^{\eta })<\infty$, for some $\eta >1$ whereas, when $\mathbb{E}({\xi }_1^{\eta })=\infty$ for some $\eta <1$, no phase transition occurs.