A note on inhomogeneous percolation on ladder graphs

Abstract: Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite connected subgraphs of $G$ and consider a model of Bernoulli bond percolation on $\mathbb{G}$ which assigns probability $q$ of being open to each edge whose projection onto $G$ lies in some subgraph of $\mathcal{C}$ and probability $p$ to every other edge. We show that the critical percolation threshold $p_{c}\left(q\right)$ is a continuous function in $\left(0,1\right)$, provided that the graphs in $\mathcal{C}$ are "well-spaced" in $G$ and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szab\'o and Valesin.