Scaling in a simple model for surface growth in a random medium

Surface growth in random media is usually governed by both the surface tension and the random local forces. Simulations on lattices mimic the former by imposing a maximum gradient m on the surface heights, and the latter by site-dependent random growth probabilities. Here we consider the limit m → ∞, where the surface grows at the site with minimal random number, independent of its neighbors. The resulting height distribution obeys a simple scaling law, which is destroyed when local surface tension is included. Our model is equivalent to Yee’s simplification of the Bak-Sneppen model for the extinction of biological species, where the height represents the number of times a biological species is exchanged.