arXiv 0711.35632007-11-22math.PRmath-phmath.MP

一类二维自破坏渗流模型$δ_c(p)$的线性下界

Linear Lower Bounds for $δ_c(p)$ for a Class of 2D Self-Destructive Percolation Models

J. van den Berg, B. N. B. de Lima

摘要：

自破坏渗流模型定义如下：考虑参数为$p > p_c$的渗流。移除无限占用群集。最后，给每个顶点（或者，对于键渗透，每个边）一个额外的机会，使其在这个阶段是空的，$δ$被占据。令$δ_c(p)$为$δ$的极小值，需要在最终配置中获得无限占用的簇。这个模型是几年前由van den Berg和Brouwer提出的。他们表明，对于正方形网格（以及其他一些满足特殊技术条件的二维网格）上的站点模型，$δ_c(p)\geq\frac{(p-p_c)}{p}$。特别地，$δ_c(p)$在$p-p_c$中至少是线性的。虽然范登伯格和布劳威尔使用的参数看起来相当严格，我们表明，他们可以适当修改，以获得类似的线性下界$δ_c(p)$（与$p$附近$p_c$）的一个更大的类的二维晶格，包括键渗流的正方形和三角形晶格，和网站渗流的星形晶格（或匹配晶格）的正方形晶格。

英文摘要：

The self-destructive percolation model is defined as follows: Consider percolation with parameter $p > p_c$. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance $δ$ to become occupied. Let $δ_c(p)$ be the minimal value of $δ$, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced some years ago by van den Berg and Brouwer. They showed that, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) that $δ_c(p)\geq\frac{(p-p_c)}{p}$. In particular, $δ_c(p)$ is at least linear in $p-p_c$. Although the arguments used by van den Berg and Brouwer look quite rigid, we show that they can be suitably modified to obtain similar linear lower bounds for $δ_c(p)$ (with $p$ near $p_c$) for a much larger class of 2D lattices, including bond percolation on the square and triangular lattices, and site percolation on the star lattice (or matching lattice) of the square lattice.

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英文摘要：

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