  In this article we study the sharpness of the phase transition for
percolation models defined on top of planar spin systems. The two examples that
we treat in detail concern the Glauber dynamics for the Ising model and a
Dynamic Bootstrap process. For both of these models we prove that their phase
transition is continuous and sharp, providing also quantitative estimates on
the two point connectivity. The techniques that we develop in this work can be
applied to a variety of different dependent percolation models and we discuss
some of the problems that can be tackled in a similar fashion. In the last
section of the paper we present a long list of open problems that would require
new ideas to be attacked.
