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Discontinuity of the percolation density in one dimensional \(1/| x- y| ^ 2\) percolation models. (English) Zbl 0613.60097
We consider one dimensional percolation models for which the occupation probability of a bond - \(K_{x,y}\), 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 \(K_{x,y}\leq p<1\) and if for long bonds \(K_{x,y}\leq \beta /| x-y|^ 2\) with \(\beta\leq 1\), regardless of how close p is to 1, ii) in models for which the above asymptotic bound holds with some \(\beta <\infty\), there is a discontinuity in the percolation density M \((\equiv P_{\infty})\) 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 \(C(\beta,p)/| x- y|^ 2.\)
The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then \(\beta M^ 2\geq 1\). This dichotomy resembles one for the magnetization in \(1/| x- y|^ 2\) Ising models which was first proposed by D. J. Thouless [Long-range order in one-dimensional Ising systems. Phys. Rev. 187, 732-733 (1969)] and further supported by the renormalization group flow equations of P. W. Anderson, G. Yuval, and D. R. Hamann [Exact results in the Kondo problem. II. ibid. B 1, 4464-4473 (1970)]. The proofs of the above percolation phenomena involve (rigorous) renormalization type arguments of a different sort.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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