We discuss several two-dimensional lattice models displaying a form of self-organized criticality. The behavior of these models is related to the phase transition of independent site percolation, which is now very well understood in two dimensions. In particular, we study the frozen percolation model (where connected components of vertices stop growing when they get too large), and we present a connection with forest-fire processes (where lightning hits independently each vertex with a small rate, and burns its entire connected component immediately).

This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and Demeter Kiss.