Probability Theory and Related Fields

, Volume 73, Issue 3, pp 369–394

The incipient infinite cluster in two-dimensional percolation

  • Harry Kesten
Article

DOI: 10.1007/BF00776239

Cite this article as:
Kesten, H. Probab. Th. Rel. Fields (1986) 73: 369. doi:10.1007/BF00776239

Summary

LetPp 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). LetPH 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)

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

     
  2. ii)

    Pp {·∣W is infinite} asppH.

     

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.

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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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