Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem

TL;DR

This paper analyzes the scaling regimes and critical dimensions of the KPZ equation with correlated noise, using a stochastic transformation to determine exponents and exploring the nature of rough phases across critical dimensions.

Contribution

It introduces a comprehensive analysis of the KPZ equation with correlated noise, deriving critical exponents to all orders and proposing a new perspective on phase differences across the critical dimension.

Findings

Critical and correction-to-scaling exponents determined to all orders.

Identification of the lower critical dimension as d_c = 2(1 + ρ).

Potential difference between rough phases above and below d_c.

Abstract

We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 \rho}) in Fourier space, as a function of \rho and the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + \rho) are genuinely different which could lead to a re-interpretation of results in the literature.