Slow convergence in bootstrap percolation

• 成果类型: Article
• 署名作者: Gravner, Janko; Holroyd, Alexander E.
• 署名单位: University of California System; University of California Davis; University of British Columbia
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP473
• 发表日期: 2008
• 页码: 909-928
• 关键词: metastability threshold; partitions; TRANSITION; INTEGRALS; models

摘要

In the bootstrap percolation model, sites in an L x 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) -> (infinity, 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 lambda = pi(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 Omega((logL)(-1/2)). In contrast, the critical window has width only Theta((logL)(-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 = 10(3000). Our results shed some light on the observed differences between simulations and rigorous asymptotics.