  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.
