Bernoulli percolation is a model for random media introduced by Broadbent and Hammersley in 1957. In this process, each vertex of a given graph is occupied or vacant, with respective probabilities p and 1-p, independently of the other vertices (for some parameter p). It is arguably one of the simplest models from statistical mechanics displaying a phase transition as the parameter p varies, i.e. a drastic change of behavior at some critical value p_c, and it has been widely studied.


Percolation can be used to analyze forest fire (or epidemics) processes. In such processes, all vertices of a lattice are initially vacant, and then become occupied at rate 1. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate, all the vertices connected to it burn immediately, i.e. they become vacant. In particular, we want to analyze the near-critical behavior of such processes, that is, when large components of occupied sites start to appear. They display a form of self-organized criticality, and the phase transition of Bernoulli percolation plays an important role: it appears "spontaneously".


This talk is based on a joint work with Rob van den Berg (CWI and VU, Amsterdam).