Slow convergence in bootstrap percolation

Abstract

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 λ=π²/18 [Probab. Theory Related Fields 125 (2003) 195–224]. 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.