Origins of scaling corrections in ballistic growth models

Abstract

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 obtained scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in divide the surface in bins of size ε and use only the maximal height inside each bin to do the statistics. Again, scaling exponents in remarkable agreement with the KPZ class were found. The binning method allowed 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.

Universality of (2+1)-dimensional restricted solid-on-solid models

“…The observation of stronger corrections for larger N s is consistent with a recent analysis of the BD [24]. This study found that corrections to scaling, for both α and β, are reduced, when the BD surface is smoothened by binning of the surface positions before analysis, thereby decreasing the height differences between neighboring sites.…”

“…1, the effective exponents suffer from stronger corrections for N > 1 than in the N = 1 case. Furthermore, our data suggest a possible oscillating convergence of β eff for N > 1, as reported in simulations of the ballistic deposition model (BD) [24]. Extrapolations based on the form (11), while in good agreement within the observed region, are prone to overfitting, where they cannot cover all possible corrections.…”

“…While in D = 1 + 1 exact solutions are known, due to the Galilean symmetry [4] and an incidental fluctuationdissipation symmetry [12], in higher dimensions KPZ has been investigated by various analytical [13][14][15][16][17][18] and numerical methods [19][20][21][22], but still debated issues remain. For example, there is a controversy on the surface growth exponents of the D = 2 + 1 KPZ, obtained by recent simulations [23,24] and a field-theoretical study [25]. Assuming that the height correlations do not exhibit multiscaling and satisfy an operator product expansion Ref.…”

Hallmarks of the Kardar–Parisi–Zhang universality class elicited by scanning probe microscopy

“…The underlying mechanism of the KPZ features is the excess velocity introduced by the probe tip scanning. While, on the one hand, it was recently shown that the surface smoothing facilitate the unveiling of the universality in genuine KPZ systems with strong correction to the scaling [64,65], on the other hand, it can introduce artificial traits in dynamics different from KPZ. We also discussed ways to rule out false positives for KPZ.…”

The probability density function tail of the Kardar–Parisi–Zhang equation in the strongly non-linear regime

“…The KPZ equation has been studied extensively, however there are some remaining controversial issues, in particular the estimates of the upper critical dimension are in the range d c = 2.8 − ∞ [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Beyond d c the critical exponents are given by the mean field theory.…”