## The Annals of Applied Probability

### First Passage Percolation for Random Colorings of $\mathbb{Z}^d$

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab. Volume 3, Number 3 (1993), 746-762.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aoap/1177005361

Digital Object Identifier
doi:10.1214/aoap/1177005361

Mathematical Reviews number (MathSciNet)
MR1233623

Zentralblatt MATH identifier
0780.60101

JSTOR

#### Citation

Fontes, Luiz; Newman, Charles M. First Passage Percolation for Random Colorings of $\mathbb{Z}^d$. Ann. Appl. Probab. 3 (1993), no. 3, 746--762. doi:10.1214/aoap/1177005361. http://projecteuclid.org/euclid.aoap/1177005361.

#### Corrections

• See Correction: L. R. G. Fontes, Charles M. Newman. Correction: First Passage Percolation for Random Colorings of $\mathbb{Z}^d$. Ann. Appl. Probab., Volume 4, Number 1 (1994), 254--254.