Coloring percolation clusters at random

Olle Häggström

Stochastic Processes and their Applications, 2001, vol. 96, issue 2, 213-242

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.

Keywords: Bond; percolation; Gibbs; measure; Quasilocality; Random-cluster; model; Positive; correlations (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (1)

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