Abstract

Bernoulli percolation can be used to analyze planar 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.

We want to analyze the behavior of such processes near and beyond criticality, that is, when large components of occupied sites appear. They display a form of self-organized criticality, where the phase transition of Bernoulli percolation plays an important role. In particular, a peculiar and striking phenomenon arises, that we call "near-critical avalanches".

This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and with Wai-Kit Lam (Taiwan University).