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Inhomogeneous percolation problems and incipient infinite clusters

1987; Institute of Physics; Volume: 20; Issue: 6 Linguagem: Inglês

10.1088/0305-4470/20/6/034

1361-6447

Jennifer Chayes, L. Chayes, R. Durrett,

Random Matrices and Applications

The authors consider inhomogeneous percolation models with density pc+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 2, the models studied suggest what sort of 'incipient objects' should be examined in random surface models.