The Annals of Applied Probability

Slow convergence in bootstrap percolation

Janko Gravner and Alexander E. Holroyd

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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.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 3 (2008), 909-928.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1211819789

Digital Object Identifier
doi:10.1214/07-AAP473

Mathematical Reviews number (MathSciNet)
MR2418233

Zentralblatt MATH identifier
1141.60062

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Bootstrap percolation cellular automaton metastability finite-size scaling crossover

Citation

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


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