Percolation: will the fire get across?

Each square holds a tree with probability p, independently of every other. Two trees belong to the same cluster if you can walk from one to the other by steps left, right, up or down, never leaving trees. A fire lit anywhere burns exactly its own cluster, so the fire’s question is the geometric question: is there a cluster that touches both sides?

Simon Broadbent and John Hammersley posed this in 1957, thinking about gas masks: whether gas can find a way through a filter whose pores are open or blocked at random.

On an infinite grid there is a sharp threshold p_c. Below it every cluster is finite; above it, exactly one cluster is infinite. For this grid (sites, four neighbours) nobody has found a formula for p_c; it is known numerically to many decimal places, $p_c \approx 0.592746$. Change the rules slightly, by blocking the links between squares instead of the squares, and the threshold becomes exactly 1/2.

On a finite grid the threshold is smeared out: crossing becomes more likely gradually as p rises. Press Measure to see by how much. The lab fills grids of 32, 64 and 128 squares a side one random square at a time, keeping track of clusters as they merge, and notes the moment one first touches both sides (the method of Newman and Ziff, 2000). The curves get steeper as the grid grows, and all three pass through about the same point near p_c. On an infinite grid the curve would be a step.

Choose At p_c and turn on Clusters. The largest cluster is full of holes of every size, and holes inside its holes: a fractal. Measured by how many trees it holds within distance R of a point, it grows like $R^{91/48}$, a dimension of about 1.896, between a line and a filled area. The number of clusters of size s falls off like $s^{-187/91}$. Both exponents are exact, and both are shared by every two-dimensional percolation problem, whatever the grid.

The fire tells the same story in time. Well above p_c it crosses in a ragged but steady front. At p_c it threads its way through a single winding path and takes far longer.

Every square here keeps its own random number and holds a tree whenever p is above it, so dragging the slider grows or thins the same forest rather than planting a new one. Choose Watch it join up and drag p slowly upwards from 0.5. Clusters grow by merging with each other, a few squares at a time, until near 0.59 the big ones fuse and one suddenly stretches from side to side.

• Oil and water flowing through rock whose pores are connected or not.
• Electrical conduction in a mixture of conducting and insulating grains: a composite switches from insulator to conductor at a threshold.
• Epidemics: if each contact passes the disease on with probability p, an outbreak can reach a large share of the population only above a threshold. The epidemic simulator shows the same switch as a reproduction number crossing 1.
• The Ising magnet: its randomly bonded clusters percolate exactly at Tc, which is how the Ising page draws them.

Real forest fires depend on wind, slope, moisture and fire jumping gaps, so this is not a model of a real fire. It is a model of connection, and the fire is a way to see it.

• Plant a new forest at p_c several times and burn each. Some get across and some do not: on a finite grid, the threshold is a matter of luck. (The grid is wider than it is tall, so a fire crossing the long way succeeds less than half the time.)
• Above p_c, clear a vertical strip with the brush. How wide does the firebreak need to be?
• Set the chart to Largest cluster and compare it with the spanning probability.
• With Clusters on, click a cluster to pick it out, then raise p a little and see what it joins.

• Ising model Up-or-down magnets that line up below one exact temperature, and lose it above.
• Clock and XY models Arrows that can point any way. No magnet forms, yet vortices pair up below T_BKT.
• Potts model q colours instead of two. Past four, the transition stops being gradual.
• Avalanches and crackling noise Disordered magnets flip in bursts of every size as the field rises.
• Hopfield memory An Ising model whose couplings store pictures, and roll a damaged one back.

New to all this? The lab’s introduction explains order parameters, critical points and how the simulations work.

• S. R. Broadbent and J. M. Hammersley (1957), Percolation processes, Math. Proc. Cambridge Philos. Soc..
• M. E. J. Newman and R. M. Ziff (2000), Efficient Monte Carlo Algorithm and High-Precision Results for Percolation, Phys. Rev. Lett.; the measuring method used here, and p_c = 0.59274621(13).
• D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor & Francis (2nd edition, 1994).