Slow Convergence in Bootstrap Percolation

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) → (∞, 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 λ = π²/18 [15]. We prove that the discrepancy between the critical parameter and its limit λ is at least Ω((log L) ^−1/2). In contrast, the critical window has width only Θ((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.