论文标题
在网站上的渗透阈值和平面图的阈值
On the site percolation threshold of circle packings and planar graphs
论文作者
论文摘要
圆形包装是磁盘的集合，平面上有不相交的内部。它自然会通过切相定义图形。结果表明，存在$ p> 0 $，因此每个圆形包装的以下内容都保留：如果每个磁盘都独立保留了概率$ p $，则有一个将磁盘连接到无穷大的途径的概率为零。使用平面图的圆形包装上得出以下结论：（i）具有参数$ p $的位点渗透性在经常性的简单平面三角形上或Benjaminii-Schramm限制有限的简单平面图上没有无限连接的组件。 （ii）用参数$ 1-p $的站点渗透在具有界限的瞬态简单平面三角形上具有无限连接的组件。这些结果为本杰米尼最近的猜想提供了支持。呈现了由磁盘以外的其他形状，在平面和更高维度中形成的图形的扩展。讨论了一些猜想和开放问题。
A circle packing is a collection of disks with disjoint interiors in the plane. It naturally defines a graph by tangency. It is shown that there exists $p>0$ such that the following holds for every circle packing: If each disk is retained with probability $p$ independently, then the probability that there is a path of retained disks connecting the origin to infinity is zero. The following conclusions are derived using results on circle packings of planar graphs: (i) Site percolation with parameter $p$ has no infinite connected component on recurrent simple plane triangulations, or on Benjamini--Schramm limits of finite simple planar graphs. (ii) Site percolation with parameter $1-p$ has an infinite connected component on transient simple plane triangulations with bounded degree. These results lend support to recent conjectures of Benjamini. Extensions to graphs formed from the packing of shapes other than disks, in the plane and in higher dimensions, are presented. Several conjectures and open questions are discussed.
