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

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.