On the critical value function in the divide and color model

Abstract

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 (\emph{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 define the critical point $r_c^G(p)$. In this paper, we mainly study the continuity properties of the function $r_c^G$, which is an instance of the question of locality for percolation. Our main result is the fact that in the case $G={\mathbb{Z}}^2$, $r_c^G$ 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 $r_c^G(p)$ as a function of $p$.

References in corpus (2)

• Positive association in the fractional fuzzy Potts model
• Confidence intervals for the critical value in the divide and color model

Cited by in corpus (3)

• Generalized Divide and Color models
• Confidence intervals for the critical value in the divide and color model
• $1$-independent percolation on ${\mathbb{Z}}^2\times K_n$