The incipient infinite cluster in twodimensional percolation
 Harry Kesten
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LetP _{ p } be the probability measure on the configurations of occupied and vacant vertices of a twodimensional graphG, under which all vertices are independently occupied (respectively vacant) with probabilityp (respectively 1p). 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:

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

P _{ p } {·∣W is infinite} asp↓p _{ H }.
On a set ofvmeasure 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.
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 Title
 The incipient infinite cluster in twodimensional percolation
 Journal

Probability Theory and Related Fields
Volume 73, Issue 3 , pp 369394
 Cover Date
 19860901
 DOI
 10.1007/BF00776239
 Print ISSN
 01788051
 Online ISSN
 14322064
 Publisher
 SpringerVerlag
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 Authors

 Harry Kesten ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Cornell University, 14853, Ithaca, NY, USA