TABLE OF CONTENTS An important extension of independent discrete percolation as considered so far, are models with dependencies between sites or edges. In this case, the state of each edge (site) can depend on the state of other edges (sites) of the graph.
We restrict ourselves here to stationary models, that is, models where the joint distribution of the state of any finite collection of edges (sites) does not change upon translations. In other words, the random graph has the same probabilistic behaviour everywhere; this should be compared with the notion of stationarity that we used in the construction of the Poisson process in Chapter 1.
The phase transition theorem generalises to these models, as long as edges (sites) that are separated by a path of minimum length k < ?, on the original infinite graph, are independent. We give a proof in the case of discrete site percolation on the square lattice. This is easily extended to the bond percolation case and to other lattice structures different from the grid. In the following, distances are taken in L 1, the so-called Manhattan distance, that is the minimum number of adjacent sites that must be traversed to connect two points.
Theorem 2.3.1 Consider an infinite square grid G, where sites can be either empty or occupied, and let k < ?. Let p be the (marginal) probability that a given site is occupied. If the states of any two sites at distance d >...
