A Note on Inhomogeneous Percolation on Ladder Graphs

Abstract

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