The Drossel–Schwabl forest-fire model

The model

Each site of a periodic square lattice is empty, a tree, or burning. An empty site grows a tree with probability $p$ each step; a tree catches fire if a neighboring site burns or lightning strikes it with probability $f$; and a burning tree becomes empty on the next step. The simulation starts with half the sites occupied by trees. Adjust $p$, $f$, or the lattice size to see how forests grow and fires spread.

Why criticality is in question

In 1992, Drossel and Schwabl proposed this model as an example of self-organized criticality: as burning, growth, and ignition become widely separated in time ($f/p\to0$) while the lattice grows, the forest would organize into clusters on all scales without tuning its density to a critical value. They proposed a distribution $N(s)\sim s^{-\tau}$ up to a cutoff $s_{\max}$, with rapid decay beyond it. Here $\tau$ describes the apparent power-law exponent; their scaling argument predicted $s_{\max}\sim\theta\ln\theta$, where $\theta=p/f$. That prediction is part of the original hypothesis, not an established two-dimensional asymptotic law.

In two dimensions, later simulations by Grassberger (2002) and Pruessner and Jensen (2002) found drifting exponents and broken simple scaling. Dense patches can produce large fires while sparse patches produce small ones: together they can look like a power law over a limited range without being scale-free. This is evidence against the original simple-scaling picture, not a proof that all asymptotic critical behavior is absent. The fitted $\tau$ and $s_{\max}$ here are exploratory; a finite lattice with correlated snapshots cannot settle the limiting question.