Approximate zero-one laws and sharpness of the percolation transition in a class of models including 2D Ising percolation

One of the most well-known classical results for site percolation on the square lattice is the equation pc + p ∗ c = 1. In words, this equation means that for all values 6= pc of the parameter p the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed ‘star’ cluster. This result is closely related to the percolation transition being sharp: Below pc the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distrubtion with an exponential tail. The analog of this result has been proved by Higuchi in 1993 for two-dimensional Ising percolation, with fixed inverse temparature β < βc, and as parameter the external field h. Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollobás and Riordan, we show that these results hold for a large class of percolation models where the vertex values can be ‘nicely’ represented (in a sense which will be defined precisely) by i.i.d. random variables. We point out that the ordinary percolation model belongs obviously to this class, and we show that also the above mentioned Ising model belongs to it. We hope that our results improve insight in the Ising percolation model, and will help to show that many other (not yet analyzed) weakly dependent percolation models also belong to the above mentioned class. ∗Part of this research has been funded by the Dutch BSIK/BRICKS project.