On the Behavior of Some Cellular Automata Related to Bootstrap Percolation
On the Behavior of Some Cellular Automata Related to Bootstrap Percolation
复制标题
关于与 Bootstrap 渗流相关的一些元胞自动机的行为
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
R. Schonmann
中科院分区：
文献类型：
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作者：
R. Schonmann
收敛到总占用率和IRC，IRC是该收敛以指数速度发生的阈值。我们找到这些临界点的所有自助渗流模型，表明它们都是0时，1 d。对于某些规则中的方向是重要的，我们表明，0 < Pc = ir- < 1，通过这些系统的定向网站渗透。最后，这些定向模型被用来获得这些模型的临界指数的估计。1.导论.相互作用粒子系统、数理统计力学和渗流等领域都从它们的相互关系中获益匪浅。在这里，我们研究了一个家庭的模型，出现在这些领域之间的接口。细胞自动机，如本文所研究的，可以被认为是相互作用的粒子系统[参见Liggett（1985）对该领域的调查]。我们所考虑的模型和渗流之间的关系将在所给出的许多证明中变得清晰，但已经可以从这些系统中的一些被称为“自举渗流”的事实中猜测出来。“最后，与统计力学的关系，虽然在本文中没有如此明确，但在例如Chalupa、Leath和赖希（1979）的文章中清楚地呈现出来，其中自举逾渗被引入与无序磁系统有关。同样在Aizenman和Lebowitz（1988）中，研究这些系统的动机来自亚稳态的（非平衡统计力学）问题。在本文中，我们主要关注的是我们模型的临界行为，即当某些参数超过某些值（临界点）时，它们的行为如何发生质的变化。像往常一样，分析这种现象的主要工具之一将是一种重整化过程，
convergence to total occupancy, and irc, the threshold for this convergence to occur exponentially fast. We locate these critical points for all the bootstrap percolation models, showing that they are both 0 when 1 d. For certain rules in which the orientation is important, we show that 0 < Pc = ir- < 1, by relating these systems to oriented site percolation. Finally, these oriented models are used to obtain an estimate for a critical exponent of these models. 1. Introduction. The fields of interacting particle systems, mathematical statistical mechanics and percolation have benefited very much from their interrelations. Here we study a family of models which has arisen in the interface among these areas. Cellular automata, such as those studied in this article, may be considered as interacting particle systems [see Liggett (1985) for a survey of this field]. The relations between the models that we consider and percolation will become clear in many of the proofs given, but can already be guessed from the fact that some of these systems are known as "bootstrap percolation." Finally, relations with statistical mechanics, while not so explicit in this article, were clearly present, for instance, in the article by Chalupa, Leath and Reich (1979), where bootstrap percolation was introduced in connection to disordered magnetic systems. Also in Aizenman and Lebowitz (1988) the motivation for studying these systems came from the (nonequilibrium statistical mechanics) problem of metastability. Our main concern in this article will be with the critical behavior of our models, that is, how their behavior changes qualitatively as some parameters cross certain values (critical points). As usual, one of the main tools in the analysis of such phenomena will be a sort of renormalization procedure by