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Percolation pp 186-235 | Cite as

Bond Percolation in Two Dimensions

  • Geoffrey Grimmett

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 ℒd 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 ℒd. Each edge of ℒ corresponds to a unique edge of ℒd, so that the percolation process on ℒ generates a percolation process on ℒd. 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 ℒd; such observations may be used to show that, in certain circumstances, ℒ contains an infinite open cluster if and only if ℒd 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 (ℒd) 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.

Keywords

Open Circuit Central Limit Theorem Open Cluster Closed Circuit Closed Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Geoffrey Grimmett
    • 1
  1. 1.School of MathematicsUniversity of BristolBristolEngland

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