OAI.Percolation.BenjaminiSchramm.cayley_main

The theorem states that, for a group Λ with a finite generating set S that is closed under inverses, does not contain the identity, and generates Λ as a group, if Λ is not Følner-amenable, then the bond-percolation model on its Cayley graph (vertices are group elements, edges join x and x·s for s in S) satisfies three conclusions. Here Følner-amenable means that for every finite K and every ε>0 there is a nonempty finite A with |(A·k) \ A| < ε|A| for all k in K. First, the two-point function of Bernoulli bond percolation at the critical parameter pc, namely the probability that x and y are joined by an open path, is the matrix of a bounded operator on ℓ²(Λ). Second, the MainConclusion property holds: the supremum of the operator norms of the two-point kernel over p<pc equals its norm at pc, this norm is finite, pc < ptwo ≤ pu, where ptwo is the supremum of parameters at which the kernel is a bounded operator and pu is the uniqueness threshold, and there exist p₁<p₂<1 with pc<p₁ such that, almost surely for independent uniform edge labels, for every p in [p₁,p₂] the subgraph of edges with label at most p has infinitely many infinite clusters. Third, for every p with pc < p < pu, Bernoulli(p) percolation almost surely has infinitely many infinite clusters.