Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

Abstract

We investigate the Kardar–Parisi–Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long–range correlated noise — characterized by its second moment R(x− x′) ∝ |x−x ′|2ρ−d — by means of dynamic field theory and the renormalization group. Using a stochastic Cole–Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension dc = 2(1 + ρ). Below the lower critical dimension, there is a line ρ∗(d) marking the stability boundary between the short-range and long-range noise fixed points. For ρ ≥ ρ∗(d), the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above ρ∗(d), one has to rely on some perturbational techniques. We discuss the location of this stability boundary ρ∗(d) in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively.