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Zbl 0613.60097
Aizenman, M.; Newman, C.M.
Discontinuity of the percolation density in one dimensional $1/\vert x- y\vert \sp 2$ percolation models. (English)
[J] Commun. Math. Phys. 107, 611-647 (1986). ISSN 0010-3616; ISSN 1432-0916/e

We consider one dimensional percolation models for which the occupation probability of a bond - $K\sb{x,y}$, 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 $K\sb{x,y}\le p<1$ and if for long bonds $K\sb{x,y}\le \beta /\vert x-y\vert\sp 2$ with $\beta\le 1$, 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 $(\equiv P\sb{\infty})$ 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 $C(\beta,p)/\vert x- y\vert\sp 2.$ \par The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then $\beta M\sp 2\ge 1$. This dichotomy resembles one for the magnetization in $1/\vert x- y\vert\sp 2$ Ising models which was first proposed by {\it 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 {\it P. W. Anderson}, {\it G. Yuval}, and {\it 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 2000:: *60K35 Interacting random processes
82B43 Percolation

Keywords: percolation models; discontinuity in the percolation density; percolation threshold; Ising models

Cited in: Zbl 1114.82015

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Zentralblatt MATH Berlin [Germany]