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 $R(\vec{x}-\vec{x}') \propto |\vec{x}-\vec{x}'|^{2ρ-d}$ --- by means of dynamic field theory and the renormalization group. Using a stochastic Cole--Hopf transformation we derive {\em exact} exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension $d_c = 2 (1+ρ)$. 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 $ρ\geq ρ_*(d)$, 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.