Discontinuity of the percolation density in one dimensional 1/|x−y|² percolation models

Abstract

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 bondsK _x,y ≦p<1 and if for long bondsK _x,y ≦β/|x−y|² with β≦1, regardless of how closep is to 1, ii) in models for which the above asymptotic bound holds with some β<∞, there is a discontinuity in the percolation densityM (≡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 asC(β,p)/|x−y|². The first two statements are consequences of a criterion which states that if the percolation densityM does not vanish then βM 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.