Forest Fire Simulator

Hold the seed at 7 and walk the density from 0.57 up to 0.62. The fire takes 7.6% of the forest, then 87%. There is no setting in between where it half-crosses. Now change the seed and leave the density alone: at 0.59 the same forest burns 7.6%, 58.8% or 19.3% depending on which random map you drew. The threshold is a statement about the average of many maps, and a single run near it tells you very little, which is what the seed box and the sweep button are for. This is also why a firebreak earns its cost. Clearing a few metres of ground breaks the connections without touching most of the fuel, and that can beat thinning a whole hillside.

Set the density to 0.55 and run it calm: the fire finds a patch, takes about 5% of the forest and stops. Give it an eastward wind of 2. Same map, same seed, and it burns 97%. No trees were added. What changed is reach, because an ember now crosses up to three cells downwind instead of one, so the gaps that used to break the cluster stop mattering. It is worth keeping this apart from directed percolation, which sounds like the same idea and does the reverse: forcing the fire to travel downwind only would take paths away, and its site threshold sits near 0.705 rather than 0.5927. Wind on this page hands the fire extra paths, so the density it needs goes down instead.

The threshold is the part that transfers, not the numbers attached to it. Oil working through sandstone, current through a random composite, an epidemic reaching R₀ = 1: each has a point below which nothing spans the system and above which something does, and that shape is the same argument every time. The exponents are not shared. The spanning cluster here grows with β = 5/36, a mean-field epidemic grows with β = 1, and conductivity near the threshold answers to a different exponent again, t, because carrying current through a cluster is a different question from belonging to one. Universality classes are narrower than the analogy that reaches for them.