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

Abstract

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.

The High Temperature Ising Model on the Triangular Lattice is a Critical Bernoulli Percolation Model

“…Thus, Theorem 1 implies that the critical point of the model coincides with its self-dual point. This is in accordance with a very natural principle which is believed to be valid in great generality, but which has been verified only in a handful of cases, including bond percolation on the square lattice [25], site percolation (see [26]) and the Divide and Colour (DaC) model [3] on the triangular lattice, and Voronoi percolation [6]. The same principle should apply to other interesting models, such as the random-cluster model (see [16]), other DaC models (see [3], Conjecture 1.7) and "confetti percolation" (see Problem 5 in [4]).…”

“…appears in several places; see, e.g., [3,23,31,32]. A renormalisation group analysis of the percolation properties of the Ising model on T, supporting the conjecture, can be found, for instance, in [27] and in Sect.…”

“…Combining Theorem 1 with results in [3] (see the end of Sect. 5 below), we next obtain the percolation phase diagram in the whole high temperature regime (β < β c ), which is qualitatively the same as for Bernoulli (independent) site percolation on T (which corresponds to the special case β = 0 of our model).…”

“…We denote by Θ(β, r) the P β,r -probability that a given vertex of the triangular lattice is contained in an infinite (+)-cluster, and define r c (β) = sup{r : Θ(β, r) = 0}. It follows from Proposition 1.8 of [3] that for all β < β c , we have r c (β) ≥ 1/2. The main result of this paper is the following theorem.…”

“…If one considers the percolation properties of spin clusters, the high temperature (β < β c ) Ising model on the triangular lattice T with no external field shows critical behaviour in the sense that the mean spin cluster size is divergent (see, e.g., [3]) and the probability that two spins are in the same spin cluster has a power law decay (see [17]). The relevant universality class is conjectured to be that of two-dimensional Bernoulli (independent) percolation.…”