Steady-state skewness and kurtosis from renormalized cumulants in $(2+1)$-dimensional stochastic surface growth

TL;DR

This paper derives and calculates higher order cumulants of surface height fluctuations in the (2+1)-dimensional KPZ equation, providing analytical estimates for skewness and kurtosis that align with numerical results.

Contribution

It introduces a diagrammatic scheme to derive renormalized cumulants up to fourth order in the stationary state of the (2+1)-dimensional KPZ surface growth model.

Findings

Calculated skewness S=0.2879

Calculated kurtosis Q=0.1995

Results agree with numerical estimations

Abstract

The phenomenon of stochastic growth of a surface on a two-dimensional substrate occurs in Nature in a variety of circumstances and its statistical characterization requires the study of higher order cumulants. Here, we consider the statistical cumulants of height fluctuations governed by the $(2+1)$-dimensional KPZ equation for flat geometry. We follow a diagrammatic scheme to derive the expressions for renormalized cumulants up to fourth order in the stationary state. Assuming a value for the roughness exponent from reliable numerical predictions, we calculate the second, third and fourth cumulants, yielding skewness $S=0.2879$ and kurtosis $Q=0.1995$. These values agree well with the available numerical estimations.