Confidence intervals for the critical value in the divide and color model

András Bálint, Vincent Beffara, Vincent Tassion

Published 2013-07-10Version 1

We obtain confidence intervals for the location of the percolation phase transition in H\"aggstr\"om's divide and color model on the square lattice $\mathbb{Z}^2$ and the hexagonal lattice $\mathbb{H}$. The resulting probabilistic bounds are much tighter than the best deterministic bounds up to date; they give a clear picture of the behavior of the DaC models on $\mathbb{Z}^2$ and $\mathbb{H}$ and enable a comparison with the triangular lattice $\mathbb{T}$. In particular, our numerical results suggest similarities between DaC model on these three lattices that are in line with universality considerations, but with a remarkable difference: while the critical value function $r_c(p)$ is known to be constant in the parameter $p$ for $p<p_c$ on $\mathbb{T}$ and appears to be linear on $\mathbb{Z}^2$, it is almost certainly non-linear on $\mathbb{H}$.