Leírás
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.