I should say a little bit more about the phenomenon of percolation, which according to Schroeder in “Fractals, Chaos, Power Laws” is a very common phenomenon involved in many different processes where there is a distinct, in some way chaotic, phase change near some critical “density,” where this critical density appears to not depend so much on the specific phenomenon, but rather depends on the problem geometry and number of degrees of freedom.
In a previous post I talked about the idea of forest fire propagation across a square grid, where at each grid point there is a fixed probability of existence of a tree. The fire can propagate unhindered when there is at least one tree as an immediate neighbour to any burning tree. Any tree surrounded by empty gridpoints will burn out and the fire will stop there. Therefore, in order for fire to propagate across (i.e. percolate through) from one side of the grid to the other, there must be at least one continuous cluster of adjacent trees spanning across the grid.
As discussed by Schroeder, the possibility of such a cluster existing depends on the point density, and for a square grid, the critical density where discrete, separate clusters join to span the grid is about 0.59. (Note, for a triangular grid the critical density is 0.5).
I wrote previously that the potential for a spanning cluster to exist does not depend on the size of the grid. I used the example of a 250 x 250 grid and showed the spanning cluster formed at P = 0.59. I also claimed that it would span a 2500 x 2500, or even 25000 x 25000 grid at P ~ 0.59, however I made that claim without proof, or any evidence (apart from the convincing claim by Schroeder in his book).
So today I explored this possibility by examining a larger grid. Computing challenges limit the practical size of this exercise, but in any event I used a 750 x 1000 grid, which is 12 times larger than the previous example. Here are some results:
At P = 0.5, we see small individual clusters form. I’ve shown the 200 largest clusters. You can see there is a characteristic size; the largest cluster is not much different than the largest cluster at P = 0.50 for a 250 x 250 grid.
Now examine the result at P = 0.59:
We see that the largest cluster spans the whole grid, as it did with the 250 x 250 grid. The maximum cluster therefore has no characteristic scale – the biggest cluster will be of the same order as the size of the full model space.
We can also see self-similarity at several different scales, breaking down only when we get to the scale of the individual burning tree. It also has an upper limit, namely the size of the model space. However, if we imagine this image is only a narrow window into an infinite (or very very big anyway) grid, then there will be self-similarity across a wide range or scales, or many orders of magnitude.
More to follow…