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

Percolation phase transition on planar spin systems