Geometry dependence in linear interface growth

TL;DR

This study shows that in linear interface growth models, the height distributions are Gaussian and universal, but their covariances depend on the geometry, with flat and radial geometries exhibiting distinct statistical properties.

Contribution

It provides a comprehensive analysis of how geometry influences the statistical properties of linear interface growth models, extending the understanding beyond nonlinear classes.

Findings

Height distributions are Gaussian and universal across geometries.

Covariance shapes differ between flat and radial geometries.

Results confirm that geometry-dependent splitting applies to linear classes.

Abstract

The effect of geometry in the statistics of \textit{nonlinear} universality classes for interface growth has been widely investigated in recent years and it is well known to yield a split of them into subclasses. In this work, we investigate this for the \textit{linear} classes of Edwards-Wilkinson (EW) and of Mullins-Herring (MH) in one- and two-dimensions. From comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes, as well as numerical integrations of the growth equations on substrates of fixed size (flat geometry) or expanding linearly in time (radial geometry), we verify that the height distributions (HDs), the spatial and the temporal covariances are universal, but geometry-dependent. In fact, the HDs are always Gaussian and, when defined in terms of the so-called "KPZ ansatz" $[h \simeq v_{\infty} t + (\Gamma t)^{\beta} \chi]$, their probability density functions $P(\chi)$ have mean null, so that all their cumulants are null, except by their variances, which assume different values in the flat and radial cases. The shape of the (rescaled) covariance curves is analyzed in detail and compared with some existing analytical results for them. Overall, these results demonstrate that the splitting of such university classes is quite general, being not restricted to the nonlinear ones.