The higher-dimensional universality conjecture for bootstrap percolation update families

Let $d\geqslant 2$ and let $\mathcal{U}$ be a $d$-dimensional bootstrap percolation update family. Write $p_c(\mathbb{Z}_n^d,\mathcal{U})$ for its critical probability, and classify $\mathcal{U}$ as subcritical, critical, or supercritical according to its stable set.

Higher-dimensional universality conjecture. \begin{enumerate} \item If $\mathcal{U}$ is subcritical, then

$$\liminf_{n\rightarrow\infty}p_c(\mathbb{Z}_n^d,\mathcal{U})>0.$$

\item If $\mathcal{U}$ is critical, then there exist $r\in\{1,\dots,d-1\}$ and $\alpha\in\mathbb{Q}$ such that

$$p_c(\mathbb{Z}_n^d,\mathcal{U})=\left(\frac{1}{\log_{(r)}n}\right)^{\alpha+o(1)}.$$

\item If $\mathcal{U}$ is supercritical, then

$$p_c(\mathbb{Z}_n^d,\mathcal{U})=n^{-\Theta(1)}.$$

\end{enumerate}

This conjecture predicts that higher-dimensional update families fall into universality classes determined by their stable-set classification: subcritical models have critical probability bounded away from zero, critical models exhibit iterated-logarithmic behaviour, and supercritical models have polynomially small critical probability. The conjecture is stated for general dimensions and update families; no resolution is supplied here.