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Coloring percolation clusters at random

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  • Häggström, Olle

Abstract

We consider the random coloring of the vertices of a graph G, that arises by first performing i.i.d. bond percolation with parameter p on G, and then assigning a random color, chosen according to some prescribed probability distribution on the finite set {0,...,r-1}, to each of the connected components, independently for different components. We call this the divide and color model, and study its percolation and Gibbs (quasilocality) properties, with emphasis on the case . On , having an infinite cluster in the underlying bond percolation process turns out to be necessary and sufficient for some single color to percolate; this fails in higher dimensions. Gibbsianness of the coloring process on , holds when p is sufficiently small, but not when p is sufficiently large. For r=2, an FKG inequality is also obtained.

Suggested Citation

  • Häggström, Olle, 2001. "Coloring percolation clusters at random," Stochastic Processes and their Applications, Elsevier, vol. 96(2), pages 213-242, December.
    • Handle: RePEc:eee:spapps:v:96:y:2001:i:2:p:213-242
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    References listed on IDEAS

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    1. Häggström, Olle, 1995. "Random-cluster measures and uniform spanning trees," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 267-275, October.
    2. Maes, C. & Redig, F. & Moffaert, A. Van & Leuven, K. U., 1999. "Almost Gibbsian versus weakly Gibbsian measures," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 1-15, January.
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    Cited by:

    1. Bálint, András & Camia, Federico & Meester, Ronald, 2009. "Sharp phase transition and critical behaviour in 2D divide and colour models," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 937-965, March.

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