论文标题
平面图上的现场渗透
Site Percolation on Planar Graphs
论文作者
论文摘要
我们证明，对于具有一端的非局部，有限的，连接的，及物的平面图，该图上的任何自动形态不变位点渗透都没有完全1个无限的1群和1个无限的0群体0群集A.S.如果我们进一步假设该位点渗透是插入的，并且A.S.〜存在独特的无限0批次，那么A.S.证明是基于对簇和轮廓之间一类精致构造的接口的分析。这些结果应用于I.I.D.〜Bernoulli位点渗透的情况，这些结果求解了1996年的Benjamini和Schramm的两个猜想（猜想7和8 in \ Cite {BS96}）。
We prove that for a non-amenable, locally finite, connected, transitive, planar graph with one end, any automorphism invariant site percolation on the graph does not have exactly 1 infinite 1-cluster and exactly 1 infinite 0-cluster a.s. If we further assume that the site percolation is insertion-tolerant and a.s.~there exists a unique infinite 0-cluster, then a.s.~there are no infinite 1-clusters. The proof is based on the analysis of a class of delicately constructed interfaces between clusters and contours. Applied to the case of i.i.d.~Bernoulli site percolation on infinite, connected, locally finite, transitive, planar graphs, these results solve two conjectures of Benjamini and Schramm (Conjectures 7 and 8 in \cite{bs96}) in 1996.
