, Volume 73, Issue 3, pp 369-394

The incipient infinite cluster in two-dimensional percolation

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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 vertexw 0. 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. P pH {·∣w 0 is connected by an occupied path to the boundary of the square [-n,n]2} asn→∞,

  2. P p {·∣W is infinite} aspp H .

On a set ofv-measure one,w 0 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.

Research supported by the NSF through a grant to Cornell University