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One of the most well-known classical results for site percolation on thesquare lattice is the equation $p_c+p_c^*=1$. In words, this equation meansthat 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 infiniteclosed "star" cluster. This result is closely related to the percolationtransition being sharp: below $p_c$, the size of the open cluster of a givenvertex is not only (a.s.) finite, but has a distribution with an exponentialtail. The analog of this result has been proven by Higuchi in 1993 fortwo-dimensional Ising percolation (at fixed inverse temperature$\beta<\beta_c$) with external field $h$, the parameter of the model. Usingsharp-threshold results (approximate zero-one laws) and a modification of anRSW-like result by Bollob\'{a}s and Riordan, we show that these results holdfor 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. randomvariables. We point out that the ordinary percolation model obviously belongsto this class and we also show that the Ising model mentioned above belongs toit.

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