Article,
The incipient infinite cluster in two-dimensional percolation
Probability Theory and Related Fields, 73 (3): 369-394 (1986)
DOI: 10.1007/BF00776239
Abstract
Let Pp be the probability measure on the configurations of
occupied and vacant vertices of a two-dimensional graph N, under which
all vertices are independently occupied (respectively vacant) with probabili-
ty p (respectively l - p ) . Let p~ be the critical probability for this system
and W the occupied cluster of some fixed vertex w o. We show that for
many graphs N, such as Z 2, or its covering graph (which corresponds to
bond percolation on •2), the following two conditional probability mea-
sures converge and have the same limit, v say:
i) Pp~.lw o is connected by an occupied path to the boundary of the
square - n , n 2 as n ~ o%
ii) Pp-IW is infinite as pSp~.
On a set of v-measure one, w 0 belongs to a unique infinite occupied
cluster, l~ say. We propose that I~ be used for the "incipient infinite
cluster". Some properties of the density of ITv and its "backbone" are
derived.
