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 $\tau$ 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 \[τ\le \exp\left(\frac{π^2}{6p}-\frac{c\log(1/p)}{\sqrt p}\right)\] for some positive constant $c$, while the classical model is known to lack the logarithmic factor.

6 pages, 1 figure, minor adjustements of the presentation

References in corpus (1)

• Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules

Cited by in corpus (1)

• Locality approach to the bootstrap percolation paradox