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

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82A43 60G60: Random fields 82A68

First-passage percolation percolation random colorings Ising model


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.

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  • 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.