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 $p\to0$ 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.

Figures (1)

• Figure 1: The two growth mechanisms. Shaded rectangles are required to be occupied; the hatched one is not occupied; the black site is infected.

Theorems & Definitions (13)

• theorem 1
• proposition 1: Filling probability
• proof : Proof of Theorem \ref{['th:main']}
• definition 1: Diagonal growth
• definition 2: Sideways growth
• definition 3: Alternating growth
• lemma 1
• proof
• lemma 2: Entropic gain
• proof