Kardar-Parisi-Zhang universality class in ($2+1$) dimensions: Universal geometry-dependent distributions and finite-time corrections

The dynamical regimes of models belonging to the Kardar-Parisi-Zhang (KPZ) universality class are investigated in $d=2+1$ by extensive simulations considering flat and curved geometries. Geometry-dependent universal distributions, different from their Tracy-Widom counterpart in one dimension, were found. Distributions exhibit finite-time corrections hallmarked by a shift in the mean decaying as $t^{-\beta }$, where $\beta$ is the growth exponent. Our results support a generalization of the ansatz $h=v_{\infty }t+{\left(\Gamma t\right)}^{\beta }\chi +\eta +\zeta t^{-\beta }$ to higher dimensions, where $v_{\infty }$, $\Gamma$, $\zeta$, and $\eta$ are nonuniversal quantities whereas $\beta$ and $\chi$ are universal and the last one depends on the surface geometry. Generalized Gumbel distributions provide very good fits of the distributions in at least four orders of magnitude around the peak, which can be used for comparisons with experiments. Our numerical results call for analytical approaches and experimental realizations of the KPZ class in two-dimensional systems.

Figure 1

Main plot: Height variance against time for the RSOS and SS models. The dashed line represents the power law $t^{0.48}$. Bottom inset: Effective growth exponents against time. The horizontal line represents the accepted KPZ value in $d=2+1$. Top inset: Interface velocity against $t^{\beta -1}$.

Figure 2

Top: Determination of the mean shift $\langle \eta \rangle$ for flat and curved geometries. The dashed line is the decay $t^{-0.24}$. Bottom: Normalized cumulants $g_n^{*}=g_n\left(t\right)/g_n\left(\infty \right)$ against scaled time for the SS (left) and RSOS (right) models in a flat geometry. Lines are (scaled) linear regressions used to determine $g_n\left(\infty \right)$.

Figure 3

Height distributions for flat growth scaled to mean 1 compared with a Gumbel distribution for $m=6$ and variance $R=0.32$. The inset shows the same data in a linear scale. The growth times are $t={10}^4$ (RSOS), $t=8000$ (SS), and $t=2000$ (etching). Scaled TW distributions are included for the sake of comparison.

Figure 4

Height distributions for curved growth scaled to mean 1 compared with a Gumbel distribution for $m=9.5$ and variance $R=0.062$. The growth times are $t=1012.4$ (RSOSC), $t=4000$ (SSC), and $t=549.7$ (Eden). Inset: Scaled height distributions disregarding the shift $\langle \eta \rangle$.