Inhomogeneous percolation problems and incipient infinite clusters

The authors consider inhomogeneous percolation models with density p c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod ^{- lambda} and assuming the existence of a correlation length exponent v for the homogeneous percolation model, they prove that in d=2, the borderline value of lambda is lambda b=1/v. If lambda >1/v then, with probability one, there is no infinite cluster, while if lambda <1/v then, with positive probability, the origin is part of an infinite cluster. This result sheds some light on numerical and theoretical predictions of certain properties of incipient infinite clusters. Furthermore, for d>2, the models studied suggest what sort of 'incipient objects' should be examined in random surface models.

Duke Scholars

Author Richard Timothy Durrett Mathematics