Short length scale fluctuations in lattice growth models

Abstract

Fluctuations of interfaces produced by lattice growth models scale as those of stochastic equations at distances r much larger than the lattice constant a. However, those equations may be derived from the short range interactions through renormalization, which suggests that universal properties may also be observed in short scale fluctuations of the lattice models. We first investigate this question in interfaces with preset exact power law structure factors by expanding the autocovariance function, which is shown to scale as r2α+constant at distances as small as r∼5a (α is the roughness exponent). Next we perform numerical simulations of lattice models in five universality classes, in one and two dimensions, and calculate the autocovariance function and the fluctuation of a local average height in their growth regimes, where finite-size effects are negligible. In cases of normal roughening with α>0, those quantities also scale as affine functions of r2α in distances from a to ∼10a, in contrast with the usual expectation that such relation is applicable only in the hydrodynamic limit. In a model with super-roughening in the Mullins–Herring class, similar relation is applicable with the local roughness exponent in one and two dimensions. In cases with α≤0, the fits of those functions diverge in the zero distance limit, in the same form as the one-point fluctuations of the corresponding stochastic equations. Finally, we study competitive models with crossovers in the roughness evolution and show that, at any given time, short and long range fluctuations scale with the exponent α of the dominant universality class at that time. Thus, short range correlations at long times do not keep a memory on the short time kinetics. These results reinforce the connection between discrete and continuous growth models by showing that their short range fluctuations have related properties.