Percolation on hyperbolic graphs

2018. 05. 14. 16:15
Rényi Intézet, második emelet, új szárny
Tom Hutchcroft (Cambridge)

In Bernoulli bond percolation, we independently decide to delete or retain edges of a graph with retention probability p. On most infinite graphs, percolation undergoes a phase transition in the sense that there exists a critical parameter 0 < p_c < 1 such that below p_c there is no infinite connected component, and above p_c there is some infinite connected component.  Benjamini and Schramm (1996) conjectured that on any nonamenable transitive graph, percolation also undergoes a second phase transition from non-uniqueness to uniqueness of the infinite cluster: That is, there exists 0 < p_c < p_u \leq 1 such if p_c < p < p_u then there are infinitely many infinite clusters, while if p > p_u there is a unique infinite cluster. In this talk, I will describe a proof of this conjecture under the additional assumption that the graph in question is Gromov hyperbolic. The proof will also establish that percolation on any Gromov hyperbolic graph has mean-field critical exponents.