On the critical value function in the divide and color model

Artikel i vetenskaplig tidskrift, 2013

The divide and color model on a graph G arises by first deleting each edge of G with probability 1-p independently of each other, then coloring the resulting connected components (i.e., every vertex in the component) black or white with respective probabilities r and 1-r, independently for different components. Viewing it as a (dependent) site percolation model, one can denote the critical point r^G_c(p). In this paper, we mainly study the continuity properties of the function r^G_c, which is an instance of the question of locality for percolation. Our main result is the fact that in the case G=Z^2, r^G_c is continuous on the interval [0,1/2); we also prove continuity at p=0 for the more general class of graphs with bounded degree. We then investigate the sharpness of the bounded degree condition and the monotonicity of r^G_c(p) as a function of p.