## 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 where

_{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)

*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 ℤ^{2}is isomorphic to ℤ^{2}. Equation (9.1) implies immediately in this case that*p*_{ c }(ℤ^{2}) = 1/2, the celebrated exact calculation proved by Kesten (1980a) using arguments based on work of Harris, Russo, Seymour, and Welsh.## Keywords

Open Circuit Central Limit Theorem Open Cluster Closed Circuit Closed Path
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## Notes

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