A note on inhomogeneous percolation on ladder graphs

TL;DR

This paper studies inhomogeneous percolation on ladder graphs, proving the critical threshold's continuity under certain spacing and size conditions, extending recent results in the field.

Contribution

It generalizes previous work by establishing the continuity of the critical percolation threshold for inhomogeneous models on ladder graphs with well-spaced subgraphs.

Findings

Critical threshold $p_c(q)$ is continuous in (0,1).

Continuity holds when subgraphs are well-spaced and have bounded size.

Generalizes recent results by Szabó and Valesin.

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.