Directed animals, quadratic and rewriting systems

Details Directed animals, quadratic and rewriting systems Directed animals, quadratic and rewriting systems

2011-12-05

arxiv.org/abs/1112.0910v1

Mathematics

Combinatorics

30 pages

Scientific paper

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 in one hand, the problem of computing the generating function $\G$ of directed animals on the square lattice, counted according to the area and the perimeter, and on the other hand, the problem to find a solution to a system of quadratic equations involving unknown matrices. The matrices solution of this problem can be finite or infinite. We were unable to find finite solutions. We present some solid clues that some infinite explicit matrices, fix points of a rewriting like system are the natural solutions of this system of equations: some strong evidences are given that the problem of finding $\G$ reduces then 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.

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