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

TL;DR

This paper derives exact critical exponents and scaling functions for the KPZ equation with spatially correlated noise using field theory and renormalization group methods, revealing phase boundaries and stability conditions.

Contribution

It provides the first exact exponents and scaling functions for the KPZ equation with long-range correlated noise, extending understanding of its phase structure.

Findings

Exact exponents and scaling functions for the roughening transition.

Identification of a stability boundary between fixed points.

Conjecture of two distinct strong-coupling phases.

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 $R(\vec{x}-\vec{x}') \propto |\vec{x}-\vec{x}'|^{2\rho-d}$ --- 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 $d_c = 2 (1+\rho)$. Below the lower critical dimension, there is a line $\rho_*(d)$ marking the stability boundary between the short-range and long-range noise fixed points. For $\rho \geq \rho_*(d)$, the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above $\rho_*(d)$, one has to rely on some perturbational techniques. We discuss the location of this stability boundary $\rho_* (d)$ 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.