Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time

Abstract

Fundamental properties of an interface evolving on a domain of size L, such as its height distribution (HD) and two-point covariances, are known to assume universal but different forms depending on whether L is fixed (flat geometry) or expands linearly in time (radial growth). The interesting situation where L varies nonlinearly, however, is far less explored and it has never been tackled for two-dimensional (2D) interfaces. Here, we study discrete KPZ growth models deposited on square lattice substrates, whose (average) lateral size enlarges as L = L0 + ωt γ . Our numerical simulations reveal that the competition between the substrate expansion and the increase of the correlation length parallel to the substrate, ξ ct 1/z , gives rise to a number of interesting results. For instance, when γ < 1/z the interface becomes fully correlated, but its squared roughness, W2, keeps increasing as W2 ∼ t 2αγ , as previously observed for 1D systems. A careful analysis of this scaling, accounting for an intrinsic width on it, allows us to estimate the roughness exponent of the 2D KPZ class as α = 0.387(1), which is very accurate and robust, once it was obtained averaging the exponents for different models and growth conditions (i.e., for various γ's and ω's). In this correlated regime, the HDs and covariances are consistent with those expected for the steady-state regime of the 2D KPZ class for flat geometry. For γ ≈ 1/z, we find a family of distributions and covariances continuously interpolating between those for the steady-state and the growth regime of radial KPZ interfaces, as the ratio ω/c augments. When γ > 1/z the system stays forever in the growth regime and the HDs always converge to the same asymptotic distribution, which is the one for the radial case. The spatial covariances, on the other hand, are (γ, ω)-dependent, showing a trend towards the covariance of a random deposition in enlarging substrates as the expansion rate increases. These results considerably generalize our understanding of the height fluctuations in 2D KPZ systems, revealing a scenario very similar to the one previously found in the 1D case.