Coloring percolation clusters at random

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 G=Zd. On Z2, 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 Zd, d⩾2, holds when p is sufficiently small, but not when p is sufficiently large. For r=2, an FKG inequality is also obtained.