Sensitive bootstrap percolation second term

Abstract

In modified two-neighbour bootstrap percolation in two dimensions each site of ${\mathbb{Z}}^2$ is initially independently infected with probability p and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time τ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $p\to 0$ it holds that

$$\mathit{\tau }\le exp\left(\frac{{\mathrm{\pi }}^2}{6p}-\frac{clog(1∕p)}{\sqrt{p}}\right)$$

for some positive constant c, while the classical model is known to lack the logarithmic factor.