Inhomogeneous percolation on ladder graphs

TL;DR

This paper studies an inhomogeneous percolation model on ladder graphs, analyzing how the critical percolation threshold varies with in-column connection probability, establishing its continuity.

Contribution

It introduces a new inhomogeneous percolation model on ladder graphs and proves the continuity of the critical threshold function with respect to the in-column probability.

Findings

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

Defines inhomogeneous percolation on ladder graphs with oriented and unoriented variants

Analyzes percolation behavior with fixed vertical connection probability

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 $(0, 1)$.