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August, 1993 First Passage Percolation for Random Colorings of $\mathbb{Z}^d$
Luiz Fontes, Charles M. Newman
Ann. Appl. Probab. 3(3): 746-762 (August, 1993). DOI: 10.1214/aoap/1177005361


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.


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Luiz Fontes. Charles M. Newman. "First Passage Percolation for Random Colorings of $\mathbb{Z}^d$." Ann. Appl. Probab. 3 (3) 746 - 762, August, 1993.


Published: August, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0780.60101
MathSciNet: MR1233623
Digital Object Identifier: 10.1214/aoap/1177005361

Primary: 60K35
Secondary: 60G60 , 82A43 , 82A68

Keywords: First-passage percolation , Ising model , percolation , random colorings

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.3 • No. 3 • August, 1993
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