Probability Theory and Related Fields

, Volume 73, Issue 3, pp 369–394 | Cite as

The incipient infinite cluster in two-dimensional percolation

  • Harry Kesten
Article

Summary

LetP p be the probability measure on the configurations of occupied and vacant vertices of a two-dimensional graphG, under which all vertices are independently occupied (respectively vacant) with probabilityp (respectively 1-p). LetP H be the critical probability for this system andW the occupied cluster of some fixed vertexw0. We show that for many graphsG, such as\(\mathbb{Z}^2 \), or its covering graph (which corresponds to bond percolation on\(\mathbb{Z}^2 \)), the following two conditional probability measures converge and have the same limit,v say:

  1. i)

    P pH {·∣w0 is connected by an occupied path to the boundary of the square [-n,n]2} asn→∞,

     
  2. ii)

    P p {·∣W is infinite} aspp H .

     

On a set ofv-measure one,w0 belongs to a unique infinite occupied cluster,WW} say. We propose thatWW} be used for the “incipient infinite cluster”. Some properties of the density ofWW} and its “backbone” are derived.

Keywords

Stochastic Process Probability Measure Probability Theory Conditional Probability Mathematical Biology 
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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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Harry Kesten
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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