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:
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i)
P pH {·∣w 0 is connected by an occupied path to the boundary of the square [-n,n]2} asn→∞,
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ii)
P p {·∣W is infinite} asp↓p 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.
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Research supported by the NSF through a grant to Cornell University
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Kesten, H. The incipient infinite cluster in two-dimensional percolation. Probab. Th. Rel. Fields 73, 369–394 (1986). https://doi.org/10.1007/BF00776239
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DOI: https://doi.org/10.1007/BF00776239