  The Burton--Keane theorem for the almost-sure uniqueness of infinite clusters
is a landmark of stochastic geometry. Let $\mu$ be a translation-invariant
probability measure with the finite-energy property on the edge-set of a
$d$-dimensional lattice. The theorem states that the number $I$ of infinite
components satisfies $\mu(I\in\{0,1\})=1$. The proof is an elegant and
minimalist combination of zero--one arguments in the presence of amenability.
The method may be extended (not without difficulty) to other problems including
rigidity and entanglement percolation, as well as to the Gibbs theory of
random-cluster measures, and to the central limit theorem for random walks in
random reflecting labyrinths. It is a key assumption on the underlying graph
that the boundary/volume ratio tends to zero for large boxes, and the picture
for non-amenable graphs is quite different.
