论文标题
双曲部位渗透
Hyperbolic site percolation
论文作者
论文摘要
在正确嵌入欧几里得或双曲线平面上时，给出了几个结果，以在准传播的平面图$ g $的位置渗透中给出一些结果。如果$（g_1，g_2）$是一对匹配的对，则源自一些准发达的马赛克$ m $，则$ p_u（g_1）+p_c（g_2）= 1 $，其中$ p_c $是无限群集群存在的至关重要的可能性，而$ p_u $是$ p_u $的关键价值，对于这种独特的clusterence clusterence clusterence clusterence clusterence clusterence clusterence clusterence of。这实现并延伸到双曲机平面，对Sykes和Essam（1964）的观察，它扩展到准传输站点模型模型Benjamini和Schramm的定理（Theorem 3.8，J。Amer。Math。Soc。14（2001）14（2001）487---507），用于延伸的延伸粘结式渗透性。 因此，$ p_u（g）+p_c（g _*）= p_u（g _*）+p_c（g）= 1 $，其中$ g _*$表示$ g $的匹配图。特别是，$ p_u（g）+p_c（g）\ ge 1 $，因此，当$ g $不适时时，我们有$ p_c（g）= p_u（g）\ ge \ frac12 $。当与同一作者（“匹配晶格对的渗透临界概率”的主要结果结合使用时，随机结构。alg。2024），我们可以获得对频繁的$ g $ gu $，而严格的不平等$ p_u（g）+p_c（g）+p_c（g）>仅如果$ g $不g $ triangulation，则只有$ p_c（g）> 1 $。 一种关键技术是一种在相应的双对图上的依赖键过程来表达匹配对上平面位点渗透过程的方法。除其他事项外，此处报告的结果在准传输图的情况下对本杰米尼和施拉姆的两种猜想（猜想7和8，Electron。Comm。1（1996）71--82）的猜想。
Several results are presented for site percolation on quasi-transitive, planar graphs $G$ with one end, when properly embedded in either the Euclidean or hyperbolic plane. If $(G_1,G_2)$ is a matching pair derived from some quasi-transitive mosaic $M$, then $p_u(G_1)+p_c(G_2)=1$, where $p_c$ is the critical probability for the existence of an infinite cluster, and $p_u$ is the critical value for the existence of a unique such cluster. This fulfils and extends to the hyperbolic plane an observation of Sykes and Essam(1964), and it extends to quasi-transitive site models a theorem of Benjamini and Schramm (Theorem 3.8, J. Amer. Math. Soc. 14 (2001) 487--507) for transitive bond percolation. It follows that $p_u (G)+p_c (G_*)=p_u(G_*)+p_c(G)=1$, where $G_*$ denotes the matching graph of $G$. In particular, $p_u(G)+p_c(G)\ge 1$ and hence, when $G$ is amenable we have $p_c(G)=p_u(G) \ge \frac12$. When combined with the main result of the companion paper by the same authors ("Percolation critical probabilities of matching lattice-pairs", Random Struct. Alg. 2024), we obtain for transitive $G$ that the strict inequality $p_u(G)+p_c(G)> 1$ holds if and only if $G$ is not a triangulation. A key technique is a method for expressing a planar site percolation process on a matching pair in terms of a dependent bond process on the corresponding dual pair of graphs. Amongst other things, the results reported here answer positively two conjectures of Benjamini and Schramm (Conjectures 7 and 8, Electron. Comm. Probab. 1 (1996) 71--82) in the case of quasi-transitive graphs.
