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

Height distributions (HDs) are key quantities to uncover universality and geometry-dependence in evolving interfaces. To quantitatively characterize 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 one- and two-dimensional substrates of sizes L=const. and L∼t. Importantly, I demonstrate that with "1-point statistics," i.e., by calculating m_{n}(t) [or κ_{n}(t)] once for all NL^{d} heights together, these corrections become very weak, so that S and K attain values very close to the asymptotic ones already at short times and for small L's. However, I find that this "1-point" (1-pt) approach fails in uncovering the universality of the HDs in the steady-state regime (SSR) of systems whose average height, h[over ¯], is a fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt height evolves as h(t)=h[over ¯](t)+s_{λ}A^{1/2}L^{α}ζ+⋯-where P(ζ) is the underlying SSR HD-and the fluctuations in h[over ¯] yield S_{1-pt}∼t^{-1/2} and K_{1-pt}∼t^{-1}. Nonetheless, by analyzing P(h-h[over ¯]), the cumulants of P(ζ) can be accurately determined. I also show that different, but universal, asymptotic values for S and K (related, so, to different HDs) can be found from the "interface statistics" in the SSR. This reveals the importance of employing the various complementary approaches to reliably determine the universality class of a given system through its different HDs.