  In the bootstrap percolation model, sites in an L by L square are initially
infected independently with probability p. At subsequent steps, a healthy site
becomes infected if it has at least 2 infected neighbours. As
(L,p)->(infinity,0), the probability that the entire square is eventually
infected is known to undergo a phase transition in the parameter p log L,
occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy
between the critical parameter and its limit lambda is at least Omega((log
L)^(-1/2)). In contrast, the critical window has width only Theta((log
L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds
which imply for example that the relative discrepancy is at least 1% even when
L = 10^3000. Our results shed some light on the observed differences between
simulations and rigorous asymptotics.
