uniqueness of the infinite cluster
/ Burton-Keane: BURR-tun KEEN /
Above the critical point an infinite open cluster exists almost surely — but how many? Could there be two, or infinitely many, disjoint infinite clusters? The answer, deeply non-obvious from the local picture, is that there is exactly one. This uniqueness theorem is one of the structural pillars of percolation: it tells us the supercritical phase has a single 'ocean' of connectivity rather than several competing infinite oceans, and it underlies almost every argument about the geometry of the supercritical phase.
The theorem of Burton and Keane (1989) states: for Bernoulli percolation on Z^d (in fact for any translation-invariant, ergodic, finite-energy measure on {0,1}^E), the number N of infinite open clusters is almost surely a constant equal to 0, 1, or infinity; and the value infinity is impossible. So when an infinite cluster exists at all (p > p_c), it is unique almost surely. The proof is a beautiful counting argument and does not use independence directly, only the finite-energy property (the conditional probability that an edge is open, given all others, is bounded away from 0 and 1). The idea: if there were three or more infinite clusters, one can build 'trifurcation points' — sites where the infinite cluster branches into three infinite arms — and translation invariance forces these to have positive density; but a region of side n can contain at most order n^d trifurcations on its boundary, contradicting a positive volume density. So 0, 1, or infinitely many is forced down to 0 or 1.
Uniqueness is what makes the supercritical phase tractable: it lets one speak of THE infinite cluster, define its density theta(p), study random walk on it, and prove that the cluster looks, on large scales, like a slightly perturbed copy of the lattice. A warning about scope: finite-energy is essential. There are natural dependent models (and percolation on non-amenable graphs like trees and hyperbolic lattices) where the infinite cluster is genuinely non-unique — on a regular tree above its threshold there are infinitely many infinite clusters. Uniqueness is a feature of amenable lattices like Z^d, not a universal law.
On Z^2 just above p_c = 1/2, simulate a large box: you see one giant cluster threading the whole region and many small finite clusters, but never two large clusters that each reach all four sides without merging. On a 3-regular tree at the same density above its threshold, by contrast, distinct infinite branches never reconnect, so there are uncountably many infinite clusters.
One infinite cluster on the amenable lattice Z^d; infinitely many on a non-amenable tree.
The Burton-Keane proof needs only translation invariance, ergodicity, and finite energy — not independence. Dropping amenability (e.g. on trees or hyperbolic graphs) breaks uniqueness entirely, so the theorem is genuinely a property of Z^d.