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

TL;DR

This paper extends classical sharpness and zero-one law results from percolation theory to a broad class of models, including two-dimensional Ising percolation, using sharp-threshold techniques and RSW-like results.

Contribution

It generalizes sharpness of phase transition results to models with vertex values represented by i.i.d. variables, including Ising percolation.

Findings

Sharp threshold results apply to a wide class of models

The class includes ordinary percolation and Ising models

Phase transition sharpness is established for these models

Abstract

One of the most well-known classical results for site percolation on the square lattice is the equation $p_c+p_c^*=1$. In words, this equation means that for all values $\neq p_c$ 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 $p_c$, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in 1993 for two-dimensional Ising percolation (at fixed inverse temperature $\beta<\beta_c$) with external field $h$, the parameter of the model. Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollob\'{a}s 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 obviously belongs to this class and we also show that the Ising model mentioned above belongs to it.