Asymptotic constant and monotonicity conjecture for two-dimensional anisotropic bootstrap percolation

Let $p$ be the initial occupation probability, and let $L_{a,b}^{th}(p)$ denote the threshold length for the two-dimensional $(a,b)$ model. Assume $a<b$, and let $C_{a,b}$ denote a model-dependent constant.

Asymptotic constant conjecture. If $a<b$, then there is a constant $C_{a,b}$ such that

$$\ln L_{a,b}^{th}(p)=C_{a,b}p^{-a}\ln^2(1/p)+o\bigl(p^{-a}\ln^2(1/p)\bigr).$$

Moreover, if $b'>b$, then $C_{a,b'}>C_{a,b}$.

The paper has matching lower and upper bounds of the same order for the general $(a,b)$ model, but does not determine the sharp threshold or the exact constant. The second assertion predicts how this constant changes as the parameter $b$ increases.