Directed Animals, Quadratic Systems and Rewriting Systems

A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function GF of directed animals on the square lattice, counted according to the area and the perimeter, and the problem of solving a system of quadratic equations involving unknown matrices. We present some solid evidence that some infinite explicit matrices, the fixed points of a rewriting like system are the natural solutions to this system of equations: some strong evidence is given that the problem of finding GF reduces to the problem of finding an eigenvector to an explicit infinite matrix. Similar properties are shown for other combinatorial questions concerning directed animals, and for different lattices.