Forest fires and near-critical percolation

After a short introduction on forest fire models, I will focus on a recent joint paper with Pierre Nolin (Hong Kong), where we study a variant of these models in which the forest is an $n\times n$ box, and ignitions come from the boundary. In particular we show that, for the case without recoveries, the probability that the center of the box is burnt tends to $0$ as $n$ tends to $\infty$. This result is not intuitively obvious, it depends heavily on (bounds for) the values of certain (near-)critical exponents for Bernoulli percolation. Our paper also presents a (weaker) version of the above result for the case with recoveries, and points out that our methods give some more insight to a remarkable forest fire model in the half-plane studied by Graf around 2015.