Scale-dependent dimension in the forest fire model

The forest fire model is a reaction-diffusion model where energy, in the form of trees, is injected uniformly, and burned (dissipated) locally. We show that the spatial distribution of fires forms a geometric structure where the fractal dimension varies continuously with the length scale. In the three-dimensional model, the dimensions vary from zero to three, proportional with In(l), as the length scale increases from l similar to 1 to a correlation length l = xi. Beyond the correlation length, which diverges with the growth rate p as xi proportional to p(-2/3), the distribution becomes homogeneous; We suggest that this picture applies to the "intermediate range" of turbulence where it provides a natural interpretation of the extended scaling that has been observed at small length scales. Unexpectedly, it might also be applicable to the spatial distribution df luminous matter in the universe. In the two-dimensional version, the dimension increases to D = 1 at a length scale l similar to 1/p, where there is a crossover to homogeneity, i.e., a jump from D = 1 to D = 2.