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

TL;DR

This paper develops tight confidence intervals for the critical point in the divide and color model on various lattices, revealing lattice-dependent behaviors and supporting universality hypotheses.

Contribution

It introduces probabilistic bounds for the critical value in the divide and color model, improving upon deterministic bounds and enabling lattice comparisons.

Findings

Confidence intervals are tighter than previous bounds.

Critical value functions differ across lattices, with non-linearity on the hexagonal lattice.

Results support universality considerations with lattice-specific behaviors.

Abstract

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 $\mathbb{H}$.