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 novel geometric structure where the fractal dimension varies continuously with the length scale. In the three dimensional model, the dimensions varies from zero to three, proportional with $log(l)$, as the length scale increases from $l \sim 1$ to a correlation length $l=\xi$. Beyond the correlation length, which diverges with the growth rate $p$ as ${\xi} \propto p^{-2/3}$, 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 of luminous matter in the universe. In the two-dimensional version, the dimension increases to D=1 at a length scale $l \sim 1/p$, where there is a cross-over to homogeneity, i. e. a jump from D=1 to D=2.