A Note on Inhomogeneous Percolation on Ladder Graphs

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 p c ( 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.