Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation

In the bootstrap percolation model, sites in an $L$ by $L$ square are initially independently declared active with probability $p$. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as $p\to 0$ and $L\to \mathrm{\infty }$ simultaneously of the probability $I(L,p)$ that the entire square is eventually active. We prove that $I(L,p)\to 1$ if $\mathrm{lim\hspace{0.17em}inf}p\log L>\lambda$, and $I(L,p)\to 0$ if $\mathrm{lim\hspace{0.17em}sup}p\log L<\lambda$, where $\lambda ={\pi }^2\mathrm{/}18$. We prove the same behaviour, with the same threshold $\lambda$, for the probability $J(L,p)$ that a site is active by time $L$ in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold ${\lambda }^{\mathrm{\prime }}={\pi }^2\mathrm{/}6$. The existence of the thresholds $\lambda ,{\lambda }^{\mathrm{\prime }}$ settles a conjecture of Aizenman and Lebowitz, while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony.