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

TL;DR

This paper applies the renormalization group to a landscape erosion model, revealing it requires infinitely many couplings for renormalizability and exhibits a surface of fixed points indicating potential scaling behavior.

Contribution

It demonstrates that the landscape erosion model is multiplicatively renormalizable only with infinitely many couplings and characterizes its fixed points and scaling properties.

Findings

Model involves infinitely many beta-functions.

Existence of a two-dimensional surface of fixed points.

Potential for nonuniversal scaling behavior.

Abstract

Standard field theoretic renormalization group is applied to the model of landscape erosion introduced by R. Pastor-Satorras and D. H. Rothman [Phys. Rev. Lett. 80: 4349 (1998); J. Stat. Phys. 93: 477 (1998)] yielding 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). Despite this fact, the one-loop counterterm can be derived albeit in a closed form in terms of the certain function $V(h)$, entering the original stochastic equation, and its derivatives with respect to the height field $h$. Its Taylor expansion gives rise to the full infinite set of the one-loop renormalization constants, beta-functions and anomalous dimensions. Instead of a set of fixed points, there is a two-dimensional surface of fixed points that is likely to contain infrared attractive region(s). If that is the case, the model exhibits scaling behaviour in the infrared range. The corresponding critical exponents are nonuniversal through the dependence on the coordinates of the fixed point on the surface, but satisfy certain universal exact relations.