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Height distributions in interface growth: The role of the averaging process

To quantitatively characterize height distributions (HDs), one uses adimensional ratios of their first central moments ($m_n$) or cumulants ($κ_n$), especially the skewness $S$ and kurtosis $K$, whose accurate estimate demands an averaging over all $L^d$ points of the height profile at a given time, in translation-invariant interfaces, and over $N$ independent samples. One way of doing this is by calculating $m_n(t)$ [or $κ_n(t)$] for each sample and then carrying out an average of them for the $N$ interfaces, with $S$ and $K$ being calculated only at the end. Another approach consists in directly calculating the ratios for each interface and, then, averaging the $N$ values. It turns out, however, that $S$ and $K$ for the growth regime HDs display strong finite-size and -time effects when estimated from these "interface statistics", as already observed in some previous works and clearly shown here, through extensive simulations of several discrete growth models belonging to the EW and KPZ classes on 1D and 2D substrates of sizes $L=const.$ and $L \sim t$. Importantly, I demonstrate that with "1-point statistics'', i.e., by calculating $m_n(t)$ [or $κ_n(t)$] once for all $N L^d$ heights together, these corrections become very weak. However, I find that this "1-point'' approach fails in uncovering the universality of the HDs in the steady state regime (SSR) of systems whose average height, $\bar{h}$, is a fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt height evolves as $h(t) = \bar{h}(t) + s_λ A^{1/2} L^α ζ+ \cdots$ -- where $P(ζ)$ is the underlying SSR HD -- and the fluctuations in $\bar{h}$ yield $S_{1pt} \sim t^{-1/2}$ and $K_{1pt} \sim t^{-1}$. Nonetheless, by analyzing $P(h-\bar{h})$, the cumulants of $P(ζ)$ can be accurately determined.