Random colorings (independent or dependent) of $\mathbb{Z}^d$ give rise to dependent first-passage percolation in which the passage time along a path is the number of color changes. Under certain conditions, we prove strict positivity of the time constant (and a corresponding asymptotic shape result) by means of a theorem of Cox, Gandolfi, Griffin and Kesten about "greedy" lattice animals. Of particular interest are i.i.d. colorings and the $d = 2$ Ising model. We also apply the greedy lattice animal theorem to prove a result on the omnipresence of the infinite cluster in high density independent bond percolation.
Publié le : 1993-08-14
Classification:
First-passage percolation,
percolation,
random colorings,
Ising model,
60K35,
82A43,
60G60,
82A68
@article{1177005361,
author = {Fontes, Luiz and Newman, Charles M.},
title = {First Passage Percolation for Random Colorings of $\mathbb{Z}^d$},
journal = {Ann. Appl. Probab.},
volume = {3},
number = {4},
year = {1993},
pages = { 746-762},
language = {en},
url = {http://dml.mathdoc.fr/item/1177005361}
}
Fontes, Luiz; Newman, Charles M. First Passage Percolation for Random Colorings of $\mathbb{Z}^d$. Ann. Appl. Probab., Tome 3 (1993) no. 4, pp. 746-762. http://gdmltest.u-ga.fr/item/1177005361/