  In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected
(in each time step) if they have at least $r$ already-infected neighbours.
Motivated by its close connections to models from statistical physics, such as
the Ising model of ferromagnetism, and kinetically constrained spin models of
the liquid-glass transition, the most extensively-studied case is the
two-neighbour bootstrap process on the two-dimensional grid $[n]^2$. Around 15
years ago, in a major breakthrough, Holroyd determined the sharp threshold for
percolation in this model, and his bounds were subsequently sharpened further
by Gravner and Holroyd, and by Gravner, Holroyd and Morris.
  In this paper we strengthen the lower bound of Gravner, Holroyd and Morris by
proving that the critical probability $p_c\big( [n]^2,2 \big)$ for percolation
in the two-neighbour model on $[n]^2$ satisfies \[p_c\big( [n]^2,2 \big) =
\frac{\pi^2}{18\log n} - \frac{\Theta(1)}{(\log n)^{3/2}}\,.\] The proof of
this result requires a very precise understanding of the typical growth of a
critical droplet, and involves a number of technical innovations. We expect
these to have other applications, for example, to the study of more general
two-dimensional cellular automata, and to the $r$-neighbour process in higher
dimensions.
