Pre-percolation closure density in critical bootstrap percolation

Determine whether, for every critical bootstrap percolation model and every ε>0, the normalized closure immediately before the percolation time satisfies \(\lim_{n\to\infty}P(|[A(\tau-1)]|/n^d>\varepsilon)=0\).

Background

The main theorem proves that, for several two-dimensional critical threshold models, the closure undergoes an abrupt transition: immediately before the percolation time it contains a negligible fraction of the torus, whereas at the percolation time it equals the entire torus.

Question 1 asks for which critical bootstrap percolation models this vanishing pre-percolation closure property holds. The authors further conjecture that it holds for all two-dimensional models generated by a π/2-rotation-invariant convex neighbourhood KK, with the critical threshold rsr_s, even when the scale parameter ss is not sufficiently large.

References

For which critical boootstrap percolation models do we have that, for all \varepsilon>0, \lim_{n\to\infty}P\left(\frac{|[A(\tau-1)]|}{nd}>\varepsilon\right)=0? We conjecture that this is the case when d=2 and (K_s,r_s) is defined by a \pi/2-rotation invariant convex K and any s\ge 1 (as opposed to s large enough).

— Explosive appearance of cores and bootstrap percolation on lattices  (2501.18976 - Hartarsky et al., 31 Jan 2025) in Question 1, Section 1.3, paragraph “General critical models”