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

Abstract

Consider subcritical Bernoulli bond percolation with 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.