The concept of self-organized criticality was coined in to describe some physical systems which present a critical-like behaviour, without the need to finely tune a parameter (like for example the temperature) to a particular value. This idea has been put forward in various real-life settings (sand piles, forest fires, avalanches, neural networks, earthquakes...), with more or less controversial outcomes. On the theoretical side, the rigorous construction and analysis of a mathematical model showing self-organized criticality is a difficult problem, and even models whose definition is very simple, like the sandpile model, are not well understood.

In this talk, I will present several toy models of self-organized criticality built upon percolation and the Ising model by introducing a feedback mechanism from the configuration onto the control parameter. The study of these models, which is a nice occasion to play around near-critical phenomena, will lead us to compare with other related models and to discuss about the ingredients necessary to obtain a self-critical behaviour.