Inhomogeneous percolation on ladder graphs

Abstract

We define an inhomogeneous percolation model on "ladder graphs" obtained as direct products of an arbitrary graph $G = (V,E)$ and the set of integers $\mathbb{Z}$ (vertices are thought of as having a "vertical" component indexed by an integer). We make two natural choices for the set of edges, producing an unoriented graph $\mathbb{G}$ and an oriented graph $\vec{\mathbb{G}}$. These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite "column" are open with probability $q$, and all other edges are open with probability $p$. For all fixed $q$ one can define the critical percolation threshold $p_c(q)$. We show that this function is continuous in $(0, 1)$.