Discontinuity of the percolation density in one dimensional 1/| x− y| 2 percolation models
10.1007/BF01205489 Cite this article as: Aizenman, M. & Newman, C.M. Commun.Math. Phys. (1986) 107: 611. doi:10.1007/BF01205489 Abstract
We consider one dimensional percolation models for which the occupation probability of a bond −
K , has a slow power decay as a function of the bond's length. For independent models — and with suitable reformulations also for more general classes of models, it is shown that: i) no percolation is possible if for short bonds x,y K ≦ x,y p<1 and if for long bonds K ≦β/| x,y x− y| 2 with β≦1, regardless of how close p is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation density M (≡ P ∞) at the percolation threshold, iii) assuming also translation invariance, in the nonpercolative regime, the mean cluster size is finite and the two-point connectivity function decays there as fast as C(β, p)/| x− y| 2. The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then β M 2>=1. This dichotomy resembles one for the magnetization in 1/| x− y| 2 Ising models which was first proposed by Thouless and further supported by the renormalization group flow equations of Anderson, Yuval, and Hamann. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.
Some of the work was done at the Institut des Hautes Etudes Scientifiques, F-91440 Buressur-Yvette, France
Research supported in part by NSF grant PHY-8301493 A02, and by a John S. Guggenheim Foundation Fellowship
Research supported in part by NSF Grant MCS-8019384, and by a John S. Guggenheim Foundation Fellowship
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