Sharp-threshold conjecture for critical bootstrap percolation models

Prove that, for the critical bootstrap percolation models covered by the paper, the percolation threshold is sharp in the sense conjectured by Balogh and Bollobás.

Background

The paper studies critical results for bootstrap percolation, including the size of the closure immediately before the percolation time. Existing results for certain models rely heavily on the Aizenman–Lebowitz approach, which is unavailable for many larger-neighbourhood models.

The authors believe that their method could extend these critical results to the models considered in the paper, specifically including the conjecture that the threshold is sharp. The statement is explicitly presented as a conjecture rather than as a theorem established in the paper.

References

In the setting of Balogh and Bollobás , we expect that, for our models, the present approach will allow proving the conjecture that the threshold is sharp.

— Explosive appearance of cores and bootstrap percolation on lattices  (2501.18976 - Hartarsky et al., 31 Jan 2025) in Section 1.3, paragraph “Other `critical' results”