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

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.

23 pages, 3 figures, version accepted in J. Stat. Mech. for publication

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Cited by in corpus (1)

• Renormalized cumulants and velocity derivative skewness in Kolmogorov turbulence