Inhomogeneous Percolation on Ladder Graphs

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 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 G and an oriented graph 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 pc(q) We show that this function is continuous in (0, 1).