On the asymptotic density in a one-dimensional self-organized critical forest-fire model

Consider the following forest-fire model where the possible locations of trees are the sites of $bold$. Each site has two possible states: `vacant' or `occupied'. Vacant sites become occupied at rate $1$. At each site ignition (by lightning) occurs at ignition rate $lambda$, the parameter of the model. When a site is ignited, its occupied cluster becomes vacant instantaneously. In the literature similar models have been studied for discrete time, finite (but large) volume and finite (but large) speed at which the fire spreads out. The most interesting behaviour seems to occur when the ignition rate goes to $0$, as this allows clusters to grow very large before being hit by lightning. It has been stated by Drossel, Clar and Schwabl (1993) that then (in our notation) the density of vacant sites (in equilibrium) is of order $1 / log(1 / lambda)$. Their proof uses a `scaling ansatz' and is not rigorous. We give, for our version of the model, a rigorous and mathematically more natural proof. Our proof shows that regardless of the initial configuration, already after time of order $log(1 / lambda)$ the density is of the above mentioned order $1 / log(1 / lambda)$. We also point out how our proof can be modified for the model studied by Drossel et al.