Polluted bootstrap percolation in three dimensions
Polluted bootstrap percolation in three dimensions
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DOI：
10.1214/20-aap1588
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发表时间：
2017-06
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通讯作者：
Janko Gravner;A. Holroyd;David J Sivakoff
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作者：
Janko Gravner;A. Holroyd;David J Sivakoff
在污染自提渗流模型中，三次晶格$\mathbb{Z}^3$的顶点被独立声明为初始占用概率$p$或以概率$q$关闭。在标准的（分别是修改的）引导规则下，如果一个顶点不是关闭的，并且它有至少$3$被占用的邻居（分别是每个坐标上有一个被占用的邻居），则在后续步骤中被占用。我们研究被占用顶点的最终密度为$p，q\到0$。我们证明了对于标准规则和修正规则，如果$q \ll p^3(\log p^{-1})^{-3}$，该密度收敛于$1$。我们的主要结果是修正模型的一个具有匹配幂的互补界：存在$C$使得最终密度收敛于$0$，如果$q > Cp^3$。对于标准模型，我们在较强条件$q>Cp^2$下建立了收敛于$0$。
In the polluted bootstrap percolation model, vertices of the cubic lattice $\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\to 0$. We show that this density converges to $1$ if $q \ll p^3(\log p^{-1})^{-3}$ for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists $C$ such that the final density converges to $0$ if $q > Cp^3$. For the standard model, we establish convergence to $0$ under the stronger condition $q>Cp^2$.