## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 3, Number 3 (1993), 631-933

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

Luiz Fontes and Charles M. Newman

#### 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: 19 April 2007

**Permanent link to this document**

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

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aoap/1177005361

**Mathematical Reviews number (MathSciNet)**

MR1233623

**Zentralblatt MATH identifier**

0780.60101

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82A43 60G60: Random fields 82A68

**Keywords**

First-passage percolation percolation random colorings Ising model

#### Citation

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

#### See also

#### 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.Project Euclid: euclid.aoap/1177005211