Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions
Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions
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亚临界 $mathcal{U}$-bootstrap 渗透模型具有不平凡的相变
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Paul Smith
中科院分区：
文献类型：
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作者：
P. Balister;B. Bollobás;Michal Przykucki;Paul Smith
我们证明了$\mathbb{Z}^2 $上的经典自举渗流模型存在具有非平凡临界概率的自然推广，并且刻画了所有具有此性质的齐次、局部、单调模型.
货车进入（在情况$d=r=2$）和舍曼（对所有$d \geq r \geq 2$）证明了$r$-邻居自举渗流模型有平凡的临界概率对$\mathbb{Z}^d$的每一个参数$d \geq r \geq 2$的选择：也就是说，一个初始的密度集$p$几乎必然使$\mathbb{Z}^d$对每一个$p>0$。这些结果有效地结束了无限格点上自举渗流的研究。
最近Bollob\'as，Smith和Uzzell引入了一类广泛的渗流模型，称为$\mathcal{U}$-bootstrap渗流，其中包括$r$-neighbor bootstrap渗流作为一种特殊情况.他们将二维$\mathcal{U}$-bootstrap渗流模型分为三类-亚临界，临界和超临界-他们证明，像经典的2-neighbor bootstrap渗流，临界和超临界$\mathcal{U}$-bootstrap渗流模型在$\mathbb{Z}^2$上具有平凡的临界概率。他们留下了一个未解决的问题，即在亚临界家庭的情况下会发生什么。在本文中，我们回答了这个问题：我们证明了每个亚临界$\mathcal{U}$-bootstrap渗流模型在$\mathbb{Z}^2$上有一个非平凡的临界概率。这是新的，除了一定的'退化'子类的对称模型，可以从下面与定向网站渗流耦合。我们的研究结果重新开放的临界概率在无限晶格上的自举渗流的研究，他们允许一个问许多问题的亚临界自举渗流模型，通常是问网站或债券渗流。
We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property.
Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r \geq 2$: that is, an initial set of density $p$ almost surely percolates $\mathbb{Z}^d$ for every $p>0$. These results effectively ended the study of bootstrap percolation on infinite lattices.
Recently Bollob\'as, Smith and Uzzell introduced a broad class of percolation models called $\mathcal{U}$-bootstrap percolation, which includes $r$-neighbour bootstrap percolation as a special case. They divided two-dimensional $\mathcal{U}$-bootstrap percolation models into three classes -- subcritical, critical and supercritical -- and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical $\mathcal{U}$-bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^2$. They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical $\mathcal{U}$-bootstrap percolation model has a non-trivial critical probability on $\mathbb{Z}^2$. This is new except for a certain `degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.
影响因子：
1.8
作者：
Bollobás, Béla;Duminil‐Copin, Hugo;Morris, Robert;Smith, Paul
通讯作者：
Smith, Paul