Scaling, cumulant ratios, and height distribution of ballistic deposition in $3+1$ and $4+1$ dimensions

We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves et al. [Phys. Rev. E 90, 052405 (2014)] in $d=2+1$ dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for $d=3+1$ and $4+1$ dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-time corrections to the scaling, that in principle belong to the Kardar-Parisi-Zhang (KPZ) universality class in both $d=3+1$ and $4+1$. Our results constitute further evidence that the upper critical dimension of the KPZ class, if it exists, is larger than 4.

Figure 1

(a) Time evolution of the squared interface width of the BD model for both original ($\varepsilon =1$) and reconstructed surfaces in $d=3+1$. The dashed line is a power law with exponent $2\beta =0.368$. Similar behavior is observed for $d=4+1$. Effective growth exponents, ${\beta }_{\mathrm{eff}}=d(lnw)/d(lnt)$, are shown in bottom panels for (b) $d=3+1$ and (c) $4+1$. The dashed horizontal lines are the growth exponents found for other models in the KPZ class with small corrections to the scaling in the respective dimensions [8, 38].

Figure 2

Determination of nonuniversal cumulants. (a) ${\langle g^2\rangle }_c={\mathrm{\Gamma }}^{2\beta }{\langle {\chi }^2\rangle }_c$ for $d=3+1$ (open symbols) and $4+1$ (filled symbols) for BD using binned surfaces with $\varepsilon =2$, 4, and 8 from top to bottom. The lines are linear regressions used to determine ${\langle g^2\rangle }_c$. (b) Determination of $\langle g\rangle =\beta \mathrm{\Gamma }\langle \chi \rangle$ for $d=3+1$ (open symbols) and $4+1$ (filled symbols) for BD using binned surfaces with $\varepsilon =2$, 3, and 4. Dashed lines are estimates of $\langle g\rangle$.

Figure 3

Interface width analysis for BD in $d=3+1$ and $4+1$. (a) Main panel: squared interface width discounting the second cumulants of height increments for systems of sizes $L=1024$ and 228 in $d=3+1$ and $4+1$, respectively. The lines are power laws with exponents $2\beta =0.368$ and 0.290. Top inset: time evolution of the intrinsic width and second cumulant of $\delta h$ for $d=3+1$. Bottom inset: effective growth exponent analysis. (b) Main panel: squared interface width discounting the second cumulants of height increments for $d=3+1$ and different sizes. Left inset: Saturated squared interface width discounting or not the second cumulant of $\delta h$ against lattice size for $d=3+1$. Right inset: Effective roughness exponent analysis for $d=3+1$ and $4+1$. (c) Squared interface width in $d=3+1$ scaled with the exponents found in our analysis.

Figure 4

Determination of dimensionless cumulant ratios for BD in (a) $d=3+1$ and (b) $4+1$. Dashed lines represent the estimates of cumulant ratios for the RSOS model taken from Ref. [9]. The BD results were obtained using a binning parameter $\varepsilon =4$.

Figure 5

Parameter determination using the KM method [46]. Main panel: Interface growth velocity for BD in $d=3+1$ (bottom curves) and $4+1$ (top curves) are represented by open and filled symbols, respectively. We show the results for the original surface (squares) and binning parameter $\varepsilon =4$ (circles). Left inset: growth velocity against substrate slope for $d=3+1$ (open symbols) and $4+1$ (filled symbols). The velocity in $d=4+1$ is subtracted by 1 to improve visualization. Right inset: linear dependence of the velocity difference $\mathrm{\Delta }v=v_L-v_{\infty }$ with the system size according to Eq. (7).

Figure 6

Determination of the average shift $\langle \eta \rangle$ in (a) $d=3+1$ and (b) $4+1$.

Figure 7

Comparison of the probability distribution functions, Eq. (17), of the original and binned surfaces of the BD model in $d=3+1$ and $4+1$ dimensions with the RSOS model. The growth times in BD models are $t=190$ and 145 for $d=3+1$ and $4+1$, respectively.