# The incipient infinite cluster in two-dimensional percolation

## Summary

Let*P*_{ 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 probability*p* (respectively 1-*p*). Let*P*_{ H } be the critical probability for this system and*W* the occupied cluster of some fixed vertex*w*_{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:

- i)
*P*_{ pH }{·∣*w*_{0}is connected by an occupied path to the boundary of the square [-*n,n*]^{2}} as*n*→∞, - ii)
*P*_{ p }{·∣*W*is infinite} as*p*↓*p*_{ H }.

On a set of*v*-measure one,*w*_{0} belongs to a unique infinite occupied cluster,*W*W} say. We propose that*W*W} be used for the “incipient infinite cluster”. Some properties of the density of*W*W} and its “backbone” are derived.

## Keywords

Stochastic Process Probability Measure Probability Theory Conditional Probability Mathematical Biology## Preview

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