标题： Bootstrap 渗透中的缓慢收敛 显示英文标题

标题： Slow Convergence in Bootstrap Percolation

摘要： 在引导渗透模型中，L×L的正方形中的站点以概率p独立地初始感染。在后续步骤中，如果一个健康的站点至少有2个感染的邻居，则会变得感染。 当(L,p)→(∞,0)时，整个正方形最终被感染的概率已知在参数p log L上经历相变，渐近发生在λ = π²/18。 我们证明临界参数与其极限λ之间的差异至少为Ω((log L)^(-1/2))。 相比之下，临界窗口的宽度仅为Θ((log L)^(-1))。 对于所谓的修改模型，我们证明了严格的显式界限，例如，当L = 10^3000时，相对差异至少为1%。 我们的结果揭示了观察到的模拟与严格渐近之间的差异的一些原因。 显示英文摘要

摘要： 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)->(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 at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((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 = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.