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

TL;DR

This paper investigates a dependent site percolation model derived from subcritical Bernoulli bond percolation on 2D lattices, revealing phase transition properties, critical probabilities, and decay behaviors using duality and RSW theorem extensions.

Contribution

It establishes the exact relation r_c(p)+r_c^*(p)=1 for the critical probabilities on the square lattice and determines r_c(p)=1/2 on the triangular lattice, advancing understanding of dependent percolation models.

Findings

Proves r_c(p)+r_c^*(p)=1 for all subcritical p on the square lattice.

Shows r_c(p)=1/2 on the triangular lattice for all subcritical p.

Demonstrates exponential decay of cluster sizes below r_c(p) and divergence at criticality.

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.