Slow convergence in bootstrap percolation

In the bootstrap percolation model, sites in an L.L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least two infected neighbors. As (L, p).(., 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 .=.2/18 [Probab. Theory Related Fields 125 (2003) 195.224]. We prove that the discrepancy between the critical parameter and its limit . is at least .((log.L).1/2). In contrast, the critical window has width only .((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=103000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.