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

TL;DR

This paper applies the renormalization group to analyze a simple model of a random walk on a fluctuating surface, revealing different scaling regimes and deriving the spreading law of the particle cloud.

Contribution

It reformulates the stochastic model as a multiplicatively renormalizable field theory and finds exact critical dimensions for various scaling regimes.

Findings

Identifies multiple fixed points corresponding to different scaling behaviors.

Derives the spreading law for the particle cloud with an exact critical dimension.

Shows the deviation from standard random walk behavior in the model.

Abstract

The field theoretic renormalization group is applied to a simple model of random walk on a rough fluctuating surface. We consider the Fokker--Planck equation for a particle in a uniform gravitational field. The surface is modelled 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 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, large distances); all the critical dimensions are found exactly. As an example, the spreading law for particle's cloud is derived. It has the form $R^2(t)\simeq t^{2/\Delta_{\omega}}$ with the exactly known critical dimension of frequency $\Delta_{\omega}$ and, in general, differs from the standard expression $R^2(t)\simeq t$ for ordinary random walk.