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 \(p_c<p_u\) for any such graph. Our proof also yields that the triangle condition \(\nabla _{p_c}<\infty \) 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.

中文翻译：

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双曲图的渗流

我们证明，在任何无法满足的Gromov双曲准传递图上的伯努利键渗流都有一个阶段，其中存在无限多个无穷簇，这证明了在双曲性附加假设下Benjamini和Schramm（1996）的一个众所周知的猜想。换句话说，对于任何这样的图，我们表明\（p_c <p_u \）。我们的证明还得出，三角形条件\（\ nabla _ {p_c} <\ infty \）在任何这样的图上都处于临界状态，已知这意味着存在多个临界指数并采用它们的均值场值。这给出了单端组的第一族实例，所有这些族的Cayley图均被证明具有渗流的平均场临界指数。