Self-organized criticality in 2D forest fire processes
Abstract：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 of p, and it has been widely studied. Percolation can be used to analyze forest fire (or epidemics) processes. In such processes, all vertices of a (two-dimensional) 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, its entire occupied cluster burns immediately (all its vertices become vacant). In particular, we want to analyze the near-critical behavior of such processes, that is, when large connected 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).