Percolation on hyperbolic graphs

TL;DR

This paper proves that Bernoulli bond percolation on nonamenable, Gromov hyperbolic, quasi-transitive graphs exhibits multiple infinite clusters at certain probabilities, confirming a longstanding conjecture under hyperbolicity assumptions.

Contribution

It establishes the existence of multiple infinite clusters in such graphs and verifies the triangle condition at criticality, providing new examples of groups with mean-field critical exponents.

Findings

Existence of multiple infinite clusters for certain percolation probabilities.

Verification of the triangle condition at criticality.

First examples of groups with Cayley graphs having mean-field critical exponents.

Abstract

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.