The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 18, Number 3 (2008), 909-928.
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 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.
Ann. Appl. Probab., Volume 18, Number 3 (2008), 909-928.
First available in Project Euclid: 26 May 2008
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Gravner, Janko; Holroyd, Alexander E. Slow convergence in bootstrap percolation. Ann. Appl. Probab. 18 (2008), no. 3, 909--928. doi:10.1214/07-AAP473. https://projecteuclid.org/euclid.aoap/1211819789