Scale-dependent dimension in the forest fire model

The forest fire model is a reaction-diffusion model where energy, in the fo rm of trees, is injected uniformly, and burned (dissipated) locally. We sho w that the spatial distribution of fires forms a geometric structure where the fractal dimension varies continuously with the length scale. In the thr ee-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 correlat ion length l = xi. Beyond the correlation length, which diverges with the g rowth rate p as xi proportional to p(-2/3), the distribution becomes homoge neous; We suggest that this picture applies to the "intermediate range" of turbulence where it provides a natural interpretation of the extended scali ng that has been observed at small length scales. Unexpectedly, it might al so be applicable to the spatial distribution df luminous matter in the univ erse. 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.