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_mathrm{c}(q). We show that this function is continuous in (0, 1).
Highlights
In this paper we examine how the critical parameter of percolation is affected by inhomogeneities
Bond percolation on the oriented graph G defined from G = (V, E) is closely related to the contact process on G
In the continuous-time Markov dynamics infected individuals recover with rate 1 and transmit the infection to each neighbour with rate λ > 0 (“infection rate”)
Summary
In this paper we examine how the critical parameter of percolation is affected by inhomogeneities. Zhang considers an independent bond percolation model on Z2 in which edges belonging to the vertical line through the origin are open with probability q, while other edges are open with probability p. It follows from standard results in percolation theory that (0, 1). Since we are far from understanding the critical behaviour of homogeneous percolation on the more general graphs G and G we consider here, analogous results to that of Zhang are beyond the scope of our work. Concerning sensitivity of the percolation threshold to an extra parameter or inhomogeneity of the underlying model, see the theory of essential enhancements developed in [1,2]
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