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

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.