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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 ${K}_{x,y}\le p<1$ and if for long bonds ${K}_{x,y}\le \beta /{|x-y|}^{2}$ 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 $\left(\equiv {P}_{\infty }\right)$ 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 ${C\left(\beta ,p\right)/|x-y|}^{2}·$

The first two statements are consequences of a criterion which states that if the percolation density M does not vanish then $\beta {M}^{2}\ge 1$. 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 (equilibrium statistical mechanics)
##### References:
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