6 research outputs found
Average site perimeter of directed animals on the two-dimensional lattices
We introduce new combinatorial (bijective) methods that enable us to compute
the average value of three parameters of directed animals of a given area,
including the site perimeter. Our results cover directed animals of any
one-line source on the square lattice and its bounded variants, and we give
counterparts for most of them in the triangular lattices. We thus prove
conjectures by Conway and Le Borgne. The techniques used are based on Viennot's
correspondence between directed animals and heaps of pieces (or elements of a
partially commutative monoid)
Directed animals, quadratic 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 \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.Comment: 27 page
The local limit of rooted directed animals on the square lattice
We consider the local limit of finite uniformly distributed directed animals
on the square lattice viewed from the root. Two constructions of the resulting
uniform infinite directed animal are given: one as a heap of dominoes,
constructed by letting gravity act on a right-continuous random walk and one as
a Markov process, obtained by slicing the animal horizontally. We look at
geometric properties of this local limit and prove, in particular, that it
consists of a single vertex at infinitely many (random) levels. Several
martingales are found in connection with the confinement of the infinite
directed animal on the non-negative coordinates.Comment: 59 pages, 16 figure