Bond Percolation in Two Dimensions

Abstract

Until recently, percolation was a game that was played largely on the plane. There is a special reason why percolation in two dimensions is more approachable than percolation in higher dimensions. To every planar two-dimensional lattice ℒ there corresponds a ‘dual’ planar lattice ℒ_d whose edges are in one-one correspondence with the edges of ℒ; furthermore, in a natural embedding of these lattices in the plane, every finite connected subgraph of ℒ is surrounded by a circuit of ℒ_d. Each edge of ℒ corresponds to a unique edge of ℒ_d, so that the percolation process on ℒ generates a percolation process on ℒ_d. In this dual pair of processes, the origin of ℒ is in an infinite open cluster if and only if it is in the interior of no closed circuit of ℒ_d; such observations may be used to show that, in certain circumstances, ℒ contains an infinite open cluster if and only if ℒ_d contains no infinite closed cluster (almost surely), which is to say that

$$ p_c \left( L \right) + p_c \left( {L_d } \right) = 1$$

(9.1)

where p _c (ℒ) and p _c (ℒ_d) are the associated critical probabilities. We saw a similar argument in the proof of Theorem (1.10), where it was shown that the square lattice is self-dual in the sense that the dual lattice of ℤ² is isomorphic to ℤ². Equation (9.1) implies immediately in this case that p _c (ℤ²) = 1/2, the celebrated exact calculation proved by Kesten (1980a) using arguments based on work of Harris, Russo, Seymour, and Welsh.