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Consider subcritical Bernoulli bond percolation with fixed parameter p<p_c.We define a dependent site percolation model by the following procedure: foreach bond cluster, we colour all vertices in the cluster black with probabilityr and white with probability 1-r, independently of each other. On the squarelattice, defining the critical probabilities for the site model and its dual,r_c(p) and r_c^*(p) respectively, as usual, we prove that r_c(p)+r_c^*(p)=1 forall subcritical p. On the triangular lattice, where our method also works, thisleads to r_c(p)=1/2, for all subcritical p. On both lattices, we obtainexponential decay of cluster sizes below r_c(p), divergence of the mean clustersize at r_c(p), and continuity of the percolation function in r on [0,1]. Wealso discuss possible extensions of our results, and formulate some naturalconjectures. Our methods rely on duality considerations and on recentextensions of the classical RSW theorem.

Sharp phase transition and critical behaviour in 2D divide and colour models