Low density series expansions for backbone properties of percolation clusters

AbstractA graph G in which each element (vertex or edge) is assigned one of the two states ‘open’ or ‘closed’, the element e being open with probability pe independently of all other elements, is known as a percolation model. The u-υ backbone in any configuration is the subgraph consisting of all elements which belong to at least one open path from vertex u to vertex υ.The properties of this backbone in the case that G is a graph representing a crystal lattice are of central importance in physical applications. Here we develop a new technique which we call the ‘extended perimeter method’ which enables low density expansions for the expected values of backbone variables on lattice graphs to be obtained in a computationally efficient way. The method is applied to the expected number of vertices which are biconnected to a given vertex on the directed square lattice