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

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

1998-08-28

arxiv.org/abs/cond-mat/9808325v1

Eur. Phys. J. B 9 (1999) 491

Physics

Condensed Matter

Statistical Mechanics

21 pages, 15 figures

Scientific paper

10.1007/s100510050790

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.

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