There are three nonempty phases for percolation on planar hyperbolic graph

You need to know: Graph, infinite graph, planar graph, connected components of a graph, basic probability theory, almost sure convergence, bijection, infimum, notation |A| for the number of elements in any finite set A.

Background: An automorphism of graph $G=(V,E)$ is a bijection $\sigma: V \to V$ , such that pair of vertices $(u,v)$ is an edge if and only if $(\sigma(u), \sigma(v))$ is an edge. A (finite or infinite) graph G is called transitive if, for every 2 vertices u and v of G, there is an automorphism $\sigma$ such that $\sigma(u)=v$ . An infinite transitive graph $G=(V,E)$ has one end if, after deletion of any finite set of vertices (and the corresponding edges), the remaining graph has exactly one infinite connected component. A graph $G=(V,E)$ is amenable if, for every $\epsilon>0$ , there is a finite set $A \subset V$ , such that $|\partial A| \leq \epsilon |A|$ , where $\partial A$ is the set of vertices outside A connected by edge to at least one vertex in A. Transitive, nonamenable, planar graph with one end are known as “planar hyperbolic graphs”.

Let $p\in[0,1]$ . The Bernoulli(p) bond percolation on $G=(V,E)$ is a subgraph of G to which each edge of G is included independently with probability p. The Bernoulli(p) site percolation on $G=(V,E)$ is a subgraph of G to which each vertex of G is included independently with probability p, and each edge $(u,v)$ is included if and only if both vertices u and v are. For given G, let $p_c(G)$ be the infimum of all $p\in[0,1]$ such that the Bernoulli(p) (bond or site) percolation on G has an infinite connected component almost surely, and let $p_u(G)$ be the infimum of all p for which this infinite connected component is unique almost surely.

The Theorem: On 30th December 1999, Itai Benjamini and Oded Schramm submitted to the Journal of the AMS a paper in which they proved, among other results, that for any planar hyperbolic graph G we have $0 < p_c(G) < p_u(G) < 1$ , for Bernoulli bond or site percolation on G.

Short context: The Theorem proves that, as p increases from 0 to 1, the Bernoulli(p) percolation on planar hyperbolic graphs passes through three distinct nonempty phases. In the first phase, $p\in(0,p_c]$ , there are no infinite connected components; in the second phase, $p\in(p_c, p_u)$ , there are infinitely many of them; and in the third phase, $p\in[p_u, 1)$ , there is a unique infinite connected component.