Ballistic deposition patterns beneath a growing Kardar-Parisi-Zhang interface

We consider a $\left(1+1\right)$-dimensional ballistic deposition process with next-nearest-neighbor interactions, which belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The focus of our analysis is on the properties of structures appearing in the bulk of a growing aggregate: a forest of independent clusters separated by “crevices.” Competition for growth (mutual screening) between different clusters results in “thinning” of this forest, i.e., the number density $c\left(h\right)$ of clusters decreases with the height $h$ of the pattern. For the discrete stochastic equation describing the process we introduce a variational formulation similar to that used for the randomly forced continuous Burgers equation. This allows us to identify the “clusters” and crevices with minimizers and shocks in the Burgers turbulence. Capitalizing on the ideas developed for the latter process, we find that $c\left(h\right)\sim h^{-\alpha }$ with $\alpha =2/3$. We compute also scaling laws that characterize the ballistic deposition patterns in the bulk: the law of transversal fluctuations of cluster boundaries and the size distribution of clusters. It turns out that the intercluster interface is superdiffusive: the corresponding exponent is twice as large as the KPZ exponent for the surface of the aggregate. Finally we introduce a probabilistic concept of ballistic growth, dubbed the “hairy” Airy process in view of its distinctive geometric features. Its statistical properties are analyzed numerically.

Figure 1

Snapshot of an aggregate obtained within the ballistic deposition of $N=2000$ particles in a periodic box of size $L=100$ with next-nearest-neighbor interactions. Black lines trace the channels (crevices) between adjacent clusters.

Figure 2

(Color online) Numerical evidence for scaling exponents identified in the text. Note that the mean height $⟨h⟩$ of the aggregate is proportional to the number of dropping events per column, $⟨h⟩\sim t/L$. We therefore plot the quantities of interest as functions of $t/L$ rather than $h$: (a) the number density of percolating clusters (and crevices) as function of $t/L$; (b) the mean-square displacement of the boundary of a percolating cluster $⟨\Delta x^2⟩$ as function of $t/L$; (c) the probability $P\left(m\right)$ to find in a large system a cluster of mass $m$. Note that departures of numerical data from theoretical predictions appear here due to evident finite-size effects.

Figure 3

(a) An aggregate growing by sequential deposition with highlighted crevices; (b) the growing aggregate in the $\left(2+1\right)$-dimensional space time; (c) density plot of second local difference (discrete analog of second derivative) of the height, which highlights the discontinuities corresponding to shocks. Panels (a) and (c) represent front and top views, respectively, of the three-dimensional structure in panel (b).

Figure 4

(Color online) (a) Correlation between the height of cluster’s boundary and the displacement of the corresponding connected path; (b) the corresponding correlation coefficient; (c) the averaged difference between the height of cluster’s right boundary and the mean height of the BD growing interface.

Figure 5

(Color online) (a) Correlation between the heights inside the cluster (“in”) and separated by a shock (“out”); (b) the corresponding correlation coefficients.