Percolation on Hyperbolic Graphs
Percolation on Hyperbolic Graphs
复制标题
双曲图上的渗滤
DOI：
10.1007/s00039-019-00498-0
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发表时间：
2018
影响因子：
2.2
通讯作者：
Tom Hutchcroft
中科院分区：
文献类型：
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作者：
Tom Hutchcroft
我们证明了在任何不可调节的，Gromov双曲拟传递图上的Bernoulli键渗透都有一个存在无穷多个无限簇的相，验证了Benjamini和Schramm（1996）在双曲性的附加假设下的一个著名猜想。换句话说，我们证明了$$p_c<p_u$$ pc<pu对于任何这样的图。我们的证明也证明了三角形条件$$\nabla _{p_c}<\infty $$∇pc<∞在任何这样的图上都是临界的，这意味着存在几个临界指数并取它们的平均场值。这给出了第一类单端群的例子，所有这些单端群的Cayley图都被证明具有渗流的平均场临界指数。
We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-transitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that $$p_c<p_u$$pc<pu for any such graph. Our proof also yields that the triangle condition $$\nabla _{p_c}<\infty $$∇pc<∞ holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-ended groups all of whose Cayley graphs are proven to have mean-field critical exponents for percolation.
双曲和抛物线单模随机映射
DOI：
10.48550/arxiv.1612.08693
发表时间：
2016
期刊：
arXiv e-prints
影响因子：
--
作者：
Angel Omer
通讯作者：
Angel Omer