James Martin: Random graph processes with forest fires

Consider the following extension of the Erdos-Renyi random graph process; in a graph on $n$ vertices, each edge arrives at rate 1, but also each vertex is struck by lightning at rate $\lambda$, in which case all the edges in its connected component are removed. Such a "mean-field forest fire" model was introduced by Rath and Toth. For appropriate ranges of $\lambda$, the model exhibits "self-organised criticality". We investigate scaling limits, involving a multiplicative coalescent with an added "deletion" mechanism. I'll mention a few other related models, including epidemic models and "frozen percolation" processes.