Geodesic
Directed animals, quadratic systems and rewriting systems
Jean-François Marckert1
1LaBRI, CNRS, Université Bordeaux 1
The electronic journal of combinatorics, Tome 19 (2012) no. 3
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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 G 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 G 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.
DOI : 10.37236/2747
Classification : 05C30, 15A30, 05A15, 06B99
Mots-clés : directed animals, algebraic systems of matrices, rewriting systems, percolation cluster

Jean-François Marckert  1

1 LaBRI, CNRS, Université Bordeaux 1 
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     title = {Directed animals, quadratic systems and rewriting systems},
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Jean-François Marckert. Directed animals, quadratic systems and rewriting systems. The electronic journal of combinatorics, Tome 19 (2012) no. 3. doi: 10.37236/2747

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