Directed Site Percolation and Dual Filling Models

This chapter discusses directed site percolation and dual filling models. A duality theory for directed percolation has begun to develop. The dual model is not a standard percolation model, but does exhibit a threshold or critical probability. The fact that the critical probabilities of the two models sum to one was used to improve the lower bound for the square lattice directed bond percolation critical probability. Restrictions on the directedness of the edges in the percolation model are needed to establish the existence of a dual filling model. Also, two different dual filling models may exist for a given directed percolation model. This chapter presents broader view of class of directed percolation models, including models on certain non-planar graphs and models with some undirected (two-way) edges, such as the stiff model.