Math @ Duke

Publications [#243443] of Richard T. Durrett
Papers Published
 Chayes, JT; Chayes, L; Durrett, R, Inhomogeneous percolation problems and incipient infinite clusters,
Journal of Physics A: Mathematical and General, vol. 20 no. 6
(1987),
pp. 15211530, ISSN 03054470 [doi]
(last updated on 2017/12/15)
Abstract: 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.


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