  We consider a forest-fire model which, somewhat informally, is described as
follows: Each site (vertex) of the square lattice is either vacant or occupied
by a tree.Vacant sites become occupied at rate 1. Further, each site is hit by
lightningat rate lambda. This lightning instantaneously destroys (makes vacant)
the occupied cluster of the site. This model is closely related to the
Drossel-Schwabl forest-fire model, which has received much attention in the
physics literature. The most interesting behaviour seems to occur when the
lightning rate goes to zero. In the physics literature it is believed that then
the system has so-called self-organized critical behaviour.
  We let the system start with all sites vacant and study, for positive but
small lambda,the behaviour near the `critical time' tc; that is, the time after
which in the modified system without lightning an infinite occupied cluster
would emerge.
  Intuitively one might expect that if, for fixed t > tc, we let simultaneously
lambda tend to 0 and m to infinity, the probability that some tree at distance
smaller than m from O is burnt before time t goes to 1. However, we show that
under a percolation-like assumption (which we can not prove but believe to be
true) this intuition is false. We compare with the case where the square
lattice is replaced by the directed binary tree, and pose some natural open
problems.
