Amenability and percolation

Percolation is a stochastic model for many natural phenomena, traditionally defined and studied on cubical lattices $\mathbb{Z}^d$. In recent years it has been discovered that some qualitatively new features of the model can be observed if it is considered on graphs whose geometry is non-Euclidean. In particular, Grimmett and Newman noticed that if one considers percolation on the direct product of $\mathbb{Z}$ and an infinite tree, then the model exhibits two phase transitions, whereas there is only one phase transition for the percolation in $\mathbb{Z}^d,\,d\geqslant2$.
My talk will be devoted to percolation on non-amenable graphs.
The notion of amenability was introduced by John von Neumann in his study of Banach-Tarski paradox for group actions. It was believed for a long time that, for a countable group, to be amenable is the same as not to contain a free subgroup, though this was disproved, to-date there is no fully algebraic criterion of amenability. However there exist many different ways to explain what an amenable group is, one of them in terms of random walks, due to Kesten: a group is amenable if and only if the spectral radius of the Markov operator corresponding to the simple random walk on the group is equal to 1. Another probabilistic criterion of amenability is provided by an open conjecture due to Benjamini and Schramm: a group is amenable if and only if the Bernoulli i.i.d percolation on any Cayley graph has a.s. 0 or 1 infinite cluster. The unicity of infinite cluster in the supercritical phase is a classical theorem of the percolation theory on $\mathbb{Z}^d$, and the argument extends easily to all amenable groups. The best result in the opposite direction is a theorem proved in my joint work with Igor Pak (2000): every nonamenable group has infinitely many Cayley graphs for which the conjecture holds. If time permits, I'll say some words about a subsequent interesting twist of the story. Namely, this theorem was further used by Gaboriau and R.Lyons and by I.Epstein and Monod to make progress in two other old open problems about amenability.