Scaling in landscape erosion: Renormalization group analysis of a model with infinitely many couplings

Applying the standard field theory renormalization group to the model of landscape erosion introduced by Pastor-Satorras and Rothman yields unexpected results: the model is multiplicatively renormalizable only if it involves infinitely many coupling constants (i.e., the corresponding renormalization group equations involve infinitely many $\beta$-functions). We show that the one-loop counterterm can nevertheless be expressed in terms of a known function $V(h)$ in the original stochastic equation and its derivatives with respect to the height field $h$. Its Taylor expansion yields the full infinite set of the one-loop renormalization constants, $\beta$-functions, and anomalous dimensions. Instead of a set of fixed points, there arises a two-dimensional surface of fixed points that quite probably contains infrared attractive regions. If that is the case, then the model exhibits scaling behavior in the infrared range. The corresponding critical exponents turn out to be nonuniversal because they depend on the coordinates of the fixed point on the surface, but they satisfy certain universal exact relations.