Field Theoretic Renormalization Group in an Infinite-Dimensional Model of Random Surface Growth in Random Environment

TL;DR

This paper applies a field theoretic renormalization group approach to analyze the large-scale behavior of a fluctuating surface influenced by a turbulent or thermal environment, modeled via stochastic Navier-Stokes and Pavlik's equations.

Contribution

It introduces an infinite-dimensional model for surface growth in a random environment and derives explicit one-loop counterterms and fixed point structures within the renormalization group framework.

Findings

Identification of three fixed point surfaces in parameter space.

Universal critical dimensions for relevant advection surfaces.

Non-universal critical dimensions when advection is irrelevant.

Abstract

The influence of a random environment on the dynamics of a fluctuating rough surface is investigated using a field theoretic renormalization group. The environment motion is modelled by the stochastic Navier--Stokes equation, which includes both a fluid in thermal equilibrium and a turbulent fluid. The surface is described by the generalized Pavlik's stochastic equation. As a result of fulfilling the renormalizability requirement, the model necessarily involves an infinite number of coupling constants. The one-loop counterterm is derived in an explicit closed form. The corresponding renormalization group equations demonstrate the existence of three two-dimensional surfaces of fixed points in the infinite-dimensional parameter space. If the surfaces contain IR attractive regions, the problem allows for the large-scale, long-time scaling behaviour. For the first surface (advection is irrelevant) the critical dimensions of the height field $\Delta_{h}$, the response field $\Delta_{h'}$ and the frequency $\Delta_{\omega}$ are non-universal through the dependence on the effective couplings. For the other two surfaces (advection is relevant) the dimensions are universal and they are found exactly.