Origins of scaling corrections in ballistic growth models

We study the ballistic deposition and the grain deposition models on two-dimensional substrates. Using the Kardar-Parisi-Zhang (KPZ) ansatz for height fluctuations, we show that the main contribution to the intrinsic width, which causes strong corrections to the scaling, comes from the fluctuations in the height increments along deposition events. Accounting for this correction in the scaling analysis, we obtain scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in dividing the surface in bins of size $\varepsilon$ and using only the maximal height inside each bin to do the statistics. Again, scaling exponents in remarkable agreement with the KPZ class are found. The binning method allows the accurate determination of the height distributions of the ballistic models in both growth and steady-state regimes, providing the universal underlying fluctuations foreseen for KPZ class in 2 + 1 dimensions. Our results provide complete and conclusive evidences that the ballistic model belongs to the KPZ universality class in $2+1$ dimensions. Potential applications of the methods developed here, in both numerics and experiments, are discussed.

Figure 1

(a) Typical deposit of the BD model in $d=1+1$ and its height profile (for $\varepsilon =1$). (b) Height profile built using local maximal heights in boxes of size $\varepsilon =4$ for the same deposit as in (a). The vertical dashed lines indicate the separation between boxes used to construct the coarse-grained profile. Typical cross sections of (c) standard and (d) binned ($\varepsilon =2$) surfaces for the BD model in $d=2+1$, with $L=800$ and $t=1000$, are shown. Arrows indicate some narrow, deep valleys removed with the binning method.

Figure 2

(a) Growth velocity against $t^{\beta -1}$ for distinct ballistic models. The surface was built using $\varepsilon$ twice the grain or particle size. Lines indicate the same quantities for the standard surface ($\varepsilon =1$). (b) Amplitude fluctuation parameter estimated via the KPZ ansatz for BD surfaces using distinct coarse-graining parameters. Lines are linear regressions to extrapolate $\Gamma$ in the limit $t\to \infty$.

Figure 3

Determination of the intrinsic width for ballistic models using the KPZ ansatz. These curves were obtained using ${\langle {\chi }^2\rangle }_c=0.235$ and $\Gamma =57$, 3500, and 43 000 for the BD, GD2, and GD4 models, respectively.

Figure 4

Squared interface width for BD surfaces using different binning parameters. The dashed line is a power law with exponent $2{\beta }_{\text{KPZ}}$. Effective growth exponents are shown in the inset. The horizontal line represents ${\beta }_{\text{KPZ}}=0.241$ [41].

Figure 5

(a) Second-order cumulant of $\delta h$ (symbols) and ${\langle h^2\rangle }_c-{\left(\Gamma t\right)}^{2\beta }{\langle {\chi }^2\rangle }_c$ (lines) against time. (b) Third-order cumulant of $\delta h$ (symbols) and ${\langle h^3\rangle }_c-{\left(\Gamma t\right)}^{3\beta }{\langle {\chi }^3\rangle }_c$ (lines) against time.

Figure 6

(a) Squared interface width subtracted (symbols) or not (solid lines) of ${\langle {\left(\delta h\right)}^2\rangle }_c$ against time for distinct ballistic models. The dashed line is a power law with exponent $2{\beta }_{\text{KPZ}}=0.483$ [41]. (b) Effective growth exponent against time for the BD model considering (open symbols) or not (closed symbols) the intrinsic width. (c) Effective roughness exponent against size for the BD model.

Figure 7

Determination of the (a) first and (b) second cumulants of the stochastic quantity $\chi$. Solid lines are linear regressions to extrapolate $\langle \chi \rangle$ and the dashed one represents ${\langle {\chi }^2\rangle }_c=0.235$. (c) Rescaled height distributions for the BD model at deposition time $t={10}^4$. The solid line is the distribution obtained for the RSOS model [24]. Here $q^{\prime }=(h-v_{\infty }t-\langle \eta \rangle )/{\left(\Gamma t\right)}^{\beta }$.