The Annals of Applied Probability

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

Luiz Fontes and Charles M. Newman

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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

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 $\mathbb{Z}^d$. Ann. Appl. Probab. 3 (1993), no. 3, 746--762. doi:10.1214/aoap/1177005361. http://projecteuclid.org/euclid.aoap/1177005361.


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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.