Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model

Abstract

The field-theoretic renormalization group is applied to a simple model of a random walk on a rough fluctuating surface. We consider the Fokker–Planck equation for a particle in a uniform gravitational field. The surface is modeled by the generalized Edwards–Wilkinson linear stochastic equation for the height field. The full stochastic model is reformulated as a multiplicatively renormalizable field theory, which allows for the application of the standard renormalization theory. The renormalization group equations have several fixed points that correspond to possible scaling regimes in the infrared range (long times and large distances); all the critical dimensions are found exactly. As an example, the spreading law for the particle’s cloud is derived. It has the form R2(t)≃t2/Δω with the exactly known critical dimension of frequency Δω and, in general, differs from the standard expression R2(t)≃t for an ordinary random walk.