Our first aim in this chapter is to present a result of Aizenman, Kesten and Newman [1987] that, under mild conditions, above the critical probability there is a unique infinite open cluster; Burton and Keane [1989] have given a very simple and elegant proof of this result. Together with Menshikov's Theorem, this uniqueness result gives an alternative proof of the Harris–Kesten Theorem; this proof is easily adapted to determine the critical probabilities of certain other lattices. The key consequence of uniqueness is that, under a symmetry assumption, the critical probabilities for bond percolation on a planar lattice and on its dual must sum to 1. We shall prove this, along with a corresponding result for site percolation, assuming only order two symmetry. Finally, we discuss the star-delta transformation, which may be used to find the critical probabilities for certain lattices that are not self-dual.

Uniqueness of the infinite open cluster – the Aizenman–Kesten–Newman Theorem

Throughout this chapter the underlying graph ∧ will be unoriented, and of finite-type. In other words, there are finitely many equivalence classes of sites under the relation in which two sites x and y are equivalent if there is an automorphism of ∧ mapping x to y, in which case we write x ∼ y. As usual, ∧ will be infinite, locally finite, and connected.

We start this chapter with the result of Aizenman, Kesten and Newman [1987] that, in this setting, with probability 1, there is at most one infinite open cluster.