Below a critical density of trees a fire stays small, and above it one spark can cross the whole forest. Epidemics on a lattice show the same kind of threshold.
Plant a forest at random and light one tree. Sparse forests stop the fire; dense ones burn from one spark, and the change is sharp, at the percolation threshold. Below, we locate it, measure the clusters there, ask what depends on the lattice, then run an epidemic on the same grid and let the forest regrow.
A burning tree gets one chance to light each neighboring tree, with the ignition probability, then turns to ash. Wind raises the chance downwind and lowers it upwind: lit from the left edge, the windy damp preset burned about three quarters of the trees in our runs, the calm one about 3%.
Figure 1. Click a tree, or use the Ignite buttons. At 100% ignition a fire at density 40% stops in a small cluster and at 75% takes almost every tree; on this 150×150 grid the line is near 59%. With every cell a tree, it falls near 50% ignition.
Light the whole left edge: does the fire reach the right edge? Figure 2 adds trees to each forest one at a time in random order and notes when a left-to-right path first appears, so one forest answers for every density (Newman and Ziff, 2000).
Figure 2. Crossing chance on grids 16 to 256 wide (darker is larger; 4,000 forests at 16 and 32, then 2,000, 1,000, 400); the legend gives each curve's 50% point. Rescaled plots against ; spread is the gap between curves as a fraction of their width. Drag the blue line, or use the arrow keys, to set the density in Figures 1 and 3.
The curves cross near at about even odds, and the jump narrows with size, from 0.16 of density (10% to 90%) at 16 wide to 0.023 at 256, as . Here is the exponent of the correlation length, the size of the largest finite clusters, which diverges as , with exactly in two dimensions (den Nijs, 1979; Smirnov and Werner, 2001). Rescaled, the curves collapse; the least spread falls at 1.39 for this seed and 1.31 to 1.37 for three others.
Now ignore the fire and look at clusters of touching trees. Each cell of Figure 3 keeps one random number and is a tree when it is below the density, so the slider grows a single forest and you can watch clusters merge into one that spans.
Figure 3. A 256×256 forest, up-down-left-right neighbors. The largest cluster is red, the others green. Figures 4 and 5 measure this forest.
Figure 4 counts clusters by size in doubling bins, leaving out a spanning one. Near the threshold they follow a power law; at 45% they bend down as large clusters become exponentially rare. The infinite-lattice slope is −187/91 ≈ −2.05 (den Nijs, 1979; Nienhuis, 1982); this forest at 59% gives about −1.8 (100 forests, −1.69 to −1.94), the edge cutting the largest clusters short.
The spanning cluster is a fractal. A solid patch fills a box as its side squared; the critical cluster, holed at every scale, as the side to the power 91/48, which follows from Figure 4's exponent (). Above the threshold the holes close beyond the correlation length and the power climbs to 2; below, the cluster ends and it falls.
Figure 4. Clusters per unit size against size for Figure 3's forest. Dashed: slope −187/91 through the bin of sizes 8 to 15.
Figure 5. Red trees in a box centred on a red tree (up to 2,000 centres) grow as the side to a power, shown here between neighbouring box sizes, for Figure 3's forest (blue) and the same forest at (gray). Readouts fit sides 5 to 65.
The threshold belongs to the lattice, and to whether sites or links are random; the exponents do not. Figure 6 runs sites on the square grid, links on it (each open with probability , the case the epidemic below needs), and sites on the triangular lattice. Their thresholds are 0.5927 (Newman and Ziff measure 0.59274621) and exactly 1/2 for both others (Sykes and Essam, 1964; Kesten, 1980). At each threshold the largest cluster grows with slope 1.85, 1.88 and 1.86 against 91/48 ≈ 1.90.
Figure 6. First panel: crossing chance on a 64-wide patch; dashed lines at the exact thresholds. Second: mean largest cluster, in sites, at exactly the threshold fraction. Per lattice, 400 samples at widths 16 to 64, 200 at 128, 100 at 256.
Put a person on every cell and infect the one in the middle. An infected person has one step to infect each susceptible neighbor, with probability , then recovers for good: SIR on a lattice. Every pair of neighbors gets exactly one chance, so the coins could be flipped in advance: open each link with probability , and the infected are those joined to the first case by open links. The final size is one bond-percolation cluster, as Grassberger (1983) first pointed out.
So the epidemic threshold is Figure 6's bond threshold, . Ignoring geometry, each new case has at most three susceptible neighbors and would suffice; the lattice needs more because an infection keeps meeting people its earlier cases already reached.
Figure 7. Final outbreak size against , first case at the center; dots are outbreaks, the line their mean. Red dashed: ; gray: 1/3. Above 1/2 the dots split into fizzles and take-offs; larger lattices sharpen the rise.
So far someone set the density. In the Drossel-Schwabl model the forest regrows: each step empty cells sprout a tree with probability , lightning hits each tree with probability , and a struck tree burns its whole cluster. With patches thicken past the threshold between strikes, yet the average density settles near 0.39 here, well below 0.593, a patchwork of patches far below and far above the threshold (Grassberger, 2002). Patches of every size give fires of every size with nothing tuned, hence the name self-organized critical, though whether it is truly critical is still argued.
Figure 8. Growth , lightning . Density climbs between fires and crashes when a big cluster ignites. In our five-minute runs the largest fire took about 2,000 trees on the small-fire preset, about 8,000 of 10,000 cells on the large-fire one, and a third of the grid on the critical one.
Figure 9 fits the fire-size slope live: not enough fires yet. The critical preset reads 1.1 to 1.2 after a minute, 1.2 to 1.3 after five. Grassberger (2002), on grids up to 65,536 wide, finds an effective value near 1.19 (his , with the cluster-size exponent) that drifts as lightning gets rarer: a grid's number, not a universal one.
Figure 9. Fire sizes, logarithmic bins, each count divided by the number of sizes in its bin. Dashed red: least-squares fit to bins holding at least three fires.
In plain percolation, large fires need a density set near ; here slow growth and fast burning make fires of every size with nobody setting it. The next essay strips that slow drive and cascading release down to a sandpile.