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 < p c . 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, r c ( p ) and r c ∗ ( p ) respectively, as usual, we prove that r c ( p ) + r c ∗ ( p ) = 1 for all subcritical p . On the triangular lattice, where our method also works, this leads to r c ( p ) = 1 / 2 , for all subcritical p . On both lattices, we obtain exponential decay of cluster sizes below r c ( p ) , divergence of the mean cluster size at r c ( 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.