A Renormalization Scheme and Skewness of Height Fluctuations in $(1+1)$-dimensional VLDS Dynamics

TL;DR

This paper develops a renormalization approach for the $(1+1)$-dimensional VLDS equation, calculating moments and skewness of height fluctuations, confirming previous numerical predictions.

Contribution

It introduces a one-loop renormalization scheme without rescaling for the VLDS equation and computes the skewness of height fluctuations.

Findings

Skewness of height fluctuations is approximately -0.0441.

Renormalized second and third moments are obtained at large scales.

Results agree with previous numerical predictions.

Abstract

We study the $(1+1)$-dimensional Villain, Lai, and Das Sarma (VLDS) equation driven by a Gaussian white noise and implement a renormalization scheme without rescaling at one-loop order. Using a diagrammatic method, we calculate the renormalized second and third moments in the large-scale and long-time limits. The ensuing skewness value is $S=-0.0441$. This (negative) value is consistent with the numerical prediction of Das Sarma \emph{et al.} [Phys. Rev. E {\bf 53} 359 (1996)].