Maximum percolation time on higher-dimensional grids

Determine the maximum percolation time M([n]^d,r) for the d-dimensional grid [n]^d under r-neighbor bootstrap percolation when 2≤r≤2d, beyond the already known case M([n]^2,2)=3n^2/8, and characterize its behavior for the remaining values of d and r.

Background

The paper contrasts its exact q-ary hypercube result with the substantially less developed theory of maximum percolation time on grid graphs. The exact value is known for the two-dimensional grid with threshold 2, but the paper states that the behavior for other dimensions and thresholds in the range 2≤r≤2d has not been explored. This explicitly leaves the determination and characterization of M([n]d,r) for those cases unresolved.

References

What is the maximum percolation time, denoted by $M([n]d,r)$, on the $[n]d$ grid when the infection threshold $ 2 r 2d$? The value $M([n]2,2)=\frac{3n2}{8}$ was established in . However, for the other values of $d$ and $r$, the behavior of $M([n]d,r)$ has not been explored.

— Maximum Percolation Time on the q-ary Hypercube  (2503.00990 - Zhu, 2 Mar 2025) in Section “Open problems,” item 2