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

We study a natural dependent percolation model introduced by Häggström. Consider subcritical Bernoulli bond percolation with a fixed parameter p<pc. We define a dependent site percolation model by the following procedure: for each bond cluster, we colour all vertices in the cluster black with probability r and white with probability 1-r, independently of each other. On the square lattice, defining the critical probabilities for the site model and its dual, rc(p) and respectively, as usual, we prove that for all subcritical p. On the triangular lattice, where our method also works, this leads to rc(p)=1/2, for all subcritical p. On both lattices, we obtain exponential decay of cluster sizes below rc(p), divergence of the mean cluster size at rc(p), and continuity of the percolation function in r on [0,1]. We also discuss possible extensions of our results, and formulate some natural conjectures. Our methods rely on duality considerations and on recent extensions of the classical RSW theorem.