We analyze the Optimal Channel Network model for river networks using both
analytical and numerical approaches. This is a lattice model in which a
functional describing the dissipated energy is introduced and minimized in
order to find the optimal configurations. The fractal character of river
networks is reflected in the power law behaviour of various quantities
characterising the morphology of the basin. In the context of a finite size
scaling Ansatz, the exponents describing the power law behaviour are calculated
exactly and show mean field behaviour, except for two limiting values of a
parameter characterizing the dissipated energy, for which the system belongs to
different universality classes. Two modified versions of the model,
incorporating quenched disorder are considered: the first simulates
heterogeneities in the local properties of the soil, the second considers the
effects of a non-uniform rainfall. In the region of mean field behaviour, the
model is shown to be robust to both kinds of perturbations. In the two limiting
cases the random rainfall is still irrelevant, whereas the heterogeneity in the
soil properties leads to new universality classes. Results of a numerical
analysis of the model are reported that confirm and complement the theoretical
analysis of the global minimum. The statistics of the local minima are found to
more strongly resemble observational data on real rivers.Comment: 27 pages, ps-file, 11 Postscript figure
