INTERACTIVE COMPANION · HOW THINGS SPREAD · STATISTICAL PHYSICS

The Burning Lattice

A live model accompanying “Why Small Fires Are Common and Big Fires Are Not Rare Enough”

Two rules. Trees appear at random cells, lightning hits random cells, and a bolt that lands in a tree burns the whole connected patch to the ground. Nothing else. The first panel below runs that model in your browser and fits an exponent to the fire sizes as they come in. The second panel shows why the fit is done by maximum likelihood rather than by the straight line everybody draws through a log-log plot.

Model 1. The lattice, burning

Green cells hold trees, dark cells are bare ground, and a patch that is burning this instant is drawn in ember. Fire is instantaneous relative to growth, which is the standard separation of time scales: nothing grows while a fire burns. Because of that separation only the ratio of the two rates matters, and it has a name:

θ = pf = growth attempts per lightning strike.

Both sliders below set rates, and the panel prints the ratio they produce. Move them in opposite directions and watch the fires get bigger; move them together and nothing at all changes, because the model cannot tell the difference.

Tree growth rate p 0.0100 The chance that a given growth attempt happens. On its own this number does nothing at all to the statistics, which is the surprising part. Drag it and watch θ change; drag the lightning rate the same way and watch θ stay put while the distribution refuses to move. Lightning rate f 2.00e-5 Lower lightning means more growth between strikes, which means denser forest and larger fires. Start at the default, where θ = 500 and the article's main run sits. Then push lightning down one notch at a time and watch the cutoff in the right-hand panel slide right. Lattice size L 256 The article used 512. Here 256 keeps the animation smooth. Shrink it to 64 and the large fires disappear, because a fire cannot be bigger than the world it burns in. That is finite-size cutoff, visible in about ten seconds. Fit window lower edge s_min 4 The upper edge is pinned at θ. Slide the lower edge and watch the fitted exponent wander by more than its own error bar. That wandering is the real uncertainty in this measurement, and the article says so in §7.

θ = p/f: – fires: – tree density ρ: – mean fire 〈s〉: – τ (MLE): –

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Model 2. Two ways to fit the same data, one of them wrong

Here the true exponent is known, because the data are drawn from a power law we chose. The left plot shows the sample; the right shows what each estimator recovered. Maximum likelihood uses every event. The regression throws the data into histogram bins, takes logarithms, and fits a straight line through whatever survives, which is where the damage happens: the bins in the tail hold one or two events each, the empty bins vanish entirely, and taking a logarithm of a small count turns Poisson noise into a systematic pull.

True exponent τ 1.50 Set it to 1.16, near the value the forest fire model gives, then to 2.50. The regression's error grows as the exponent does, because a steeper law puts fewer events in the tail bins and the empty ones are the ones that get silently dropped. Sample size n 200,000 The default is the article's own 200,000 draws, which is where the reference figures beside Model 1 come from. Pull n down and watch the maximum likelihood error stay near zero while the raw regression falls further, from 0.53 low at 200,000 draws to over 1.0 low at a thousand: thinning the sample empties more tail bins, and empty bins are the ones that get silently dropped. More data does shrink that bias, but it never removes it, which is the thing a biased estimator does and a consistent one does not. Upper support limit s_max 10,000 This is where the power law is truncated. Widen it and the raw-histogram regression gets worse, because a longer tail means more bins holding zero or one event. Regression style Log-binning is the standard repair and it helps a lot. It still does not get back to the maximum likelihood answer, and it leaves you choosing a bin count that nobody can justify.

true τ: – MLE: – regression: – regression error: –

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Article main run: L = 512, θ = 500, 120,000 fires, seed 20251124
Fitted exponent: τ = 1.1269 ± 0.0025 over s in [4, 500]
Tree density: ρ = 0.39263 at θ = 500
Mass balance: 〈s〉 = θ(1 − ρ)/ρ, exact, no fitting
Naive regression: low by 0.004 on the fire data, by 0.53 on a known τ = 1.5

What to distrust here

The lattice is not a forest. Fire spreads to every touching tree and to nothing else, so there is no wind, no slope, no ember thrown across a road, no wet season and no species that resists burning. Trees appear at uniformly random cells, which no seed does. Burnt ground is ready for a new tree the instant the fire passes.

And the exponent you are watching is not a constant of nature. Raise θ by a factor of ten and it moves, which is the whole argument the article is about. If the readout looks stable, that is because you have been watching one value of θ for thirty seconds.