We present a comprehensive numerical investigation of non-universal parameters and corrections related to interface fluctuations of models belonging to the Kardar-Parisi-Zhang (KPZ) universality class, in d = 1 + 1, for both flat and curved geometries. We analyzed two classes of models. In the isotropic models the non-universal parameters are uniform along the surface, whereas in the anisotropic growth they vary. In the latter case, that produces curved surfaces, the statistics must be computed independently along fixed directions.where χ is a Tracy-Widom (geometry-dependent) distribution and η is a timeindependent correction, is probed.Our numerical analysis shows that the nonuniversal parameter Γ determined through the first cumulant leads to a very good accordance with the extended KPZ ansatz for all investigated models in contrast with the estimates of Γ obtained from higher order cumulants that indicate a violation of the generalized ansatz for some of the studied models. We associate the discrepancies to corrections of unknown nature, which hampers an accurate estimation of Γ at finite times. The discrepancies in Γ via different approaches are relatively small but sufficient to modify the scaling law t −1/3 that characterize the finite-time corrections due to η. Among the investigated models, we have revisited an off-lattice Eden model that supposedly disobeyed the shift in the mean scaling as t −1/3 and showed that there is a crossover to the expected regime. We have found model-dependent (non-universal) corrections for cumulants of order n ≥ 2. All investigated models are consistent with a further term of order t −1/3 in the KPZ ansatz.

We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions are obeyed by the restricted solid-on-solid (RSOS) model for substrates with dimensions up to d = 6. Analyzing different restriction conditions, we show that height distributions of the interface are universal for all investigated dimensions. It means that fluctuations are not negligible and, consequently, the system is still below the upper critical dimension at d = 6. The extrapolation of the data to dimensions d ≥ 7 predicts that the upper critical dimension of the KPZ class is infinite.PACS numbers: 68.43. Hn, 68.35.Fx, 81.15.Aa, Interface dynamics in nature is mostly a nonequilibrium process [1] and the Kardar-Parisi-Zhang (KPZ) universality class introduced by the equation [2] ∂h(x, t) ∂twhere ξ is a white noise of zero mean and amplitude √ D, is certainly one of the most relevant problems in non-equilibrium interface science [3,4].Much is known about KPZ class in d = 1 + 1 including exact solutions [5], experimental realizations [6], and fine-tuning properties as universal schema for finite-time corrections [6][7][8][9] and for the crossover to the stationary regime [10]. The compilation of all results gave rise to the extended KPZ ansatz, whose main ideas were formerly introduced by Krug et al. [11], for the interface evolution in the non-stationary regime where the height at each surface point evolves aswhere s λ = sgn(λ), β is the growth exponent, and χ is an stochastic variable, whose universal distribution depends only on the growth geometry [12]. The spatial correlations, which also depend on the growth geometry, are also known and given by the so-called Airy processes [3]. The parameters v ∞ , Γ and η are non-universal, being the last one responsible by a shift in the distribution of the scaled height,in relation to the asymptotic distributions χ. Except for the very specific case where η = 0 [7], the shift vanishes as q − χ ∼ t −β . Simulations of several models that, in principle, belong to the KPZ class have shown that the ansatz given by Eq. (2) can be extended to d = 2 + 1 with universal and geometry-dependent stochastic fluctuations [13][14][15][16] [24][25][26][27][28][29] show that KPZ upper critical dimension, if it exists, is higher than four. Particularly, in Refs. [23,28] was suggested that d u = ∞. A short but comprehensive review of the state of the art is presented in Ref. [24].Much of these discussions on d u were based on scaling exponents. The squared interface width, defined as the variance of the interface height profile, evolves according to the Family-Vicsek ansatz [30]where X n c denotes the nth cumulant of X. The scaling function f (x) = x 2β for x 1 and f (x) = const if x 1. The roughness α and dynamical z exponents obey the scaling relation α + z = 2 independent of the dimension [3], which was checked up to 5+1 dimensions [26]. For 1 t L z , W ∼ t β where the growth exponent is given by β = α/z. For d d u , we have α = β = 0 and z = 2.In this work, we investigate the ...

PACS 81.15.Aa -Theory and models of film growth PACS 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion PACS 89.75.Da -Systems obeying scaling laws Abstract. -We investigate the radius distributions (RD) of surfaces obtained with large-scale simulations of radial clusters that belong to the KPZ universality class. For all investigated models, the RDs are given by the Tracy-Widom distribution of the Gaussian unitary ensemble, in agreement with the conjecture of the KPZ universality class for curved surfaces. The quantitative agreement was also confirmed by two-point correlation functions asymptotically given by the covariance of the Airy2 process. Our simulation results fill the last lacking gap of the conjecture that had been recently verified analytically and experimentally.

We study the continuous absorbing-state phase transition in the contact process on the VoronoiDelaunay lattice. The Voronoi construction is a natural way to introduce quenched coordination disorder in lattice models. We simulate the disordered system using the quasistationary simulation method and determine its critical exponents and moment ratios. Our results suggest that the critical behavior of the disordered system is unchanged with respect to that on a regular lattice, i.e., that of directed percolation.

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 onedimension, were found. Distributions exhibit finite-time corrections hallmarked by a shift in the mean decaying as t −β , where β is the growth exponent. Our results support a generalization of the ansatz h = v∞t + (Γt) β χ + η + ζt −β to higher dimensions, where v∞, Γ, ζ and η are nonuniversal quantities whereas β and χ 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 KPZ class in two-dimensional systems. where ξ is a white noise of mean zero and amplitude √ D. Despite of its original conception for evolving interfaces, the KPZ equation has also found its place in others important physical systems [4].A great advance in the theoretical understanding of the KPZ universality class has begun at early 2000s with the seminal works of Johansson [5] and Prähofer and Spohn [6] presenting analytical asymptotic solutions of some models in the KPZ class. These solutions link the height's stochastic fluctuations to universal distributions [7] of the random matrix theory. Inspired in these exact results, the ansatzwith the exactly known growth exponent β = 1/3, was conjectured as describing the asymptotic interface fluctuations of any model belonging to KPZ class in d = 1 + 1 [4,8]. In this equation, s λ = sgn(λ), while the asymptotic velocity v ∞ and Γ are model dependent parameters and χ is a universal random variable with time-independent distribution given by the Gaussian orthogonal ensemble (GOE) for flat geometries [5,6] where η and ζ are non-universal. The correction η introduces a shift in the distribution of the scaled height q = h−v∞t s λ (Γt) β in relation to the asymptotic distributions. The hallmark of this correction, a shift in the mean vanishing as q − χ ∼ t −1/3 , has been verified in the crystal liquid experiments [2] and computer simulations of several models [9,10,16]. To our knowledge, only two exceptions have been reported. In the first one, Ferrari and Frings [15] analyzed the partially asymmetric simple exclusion process and found a specific value of the asymmetry parameter where there is no correction up order O(t −2/3 ). Off-lattice simulations of an Eden model consistent with a decay t −2/3 were reported [11], but a subsequent analysis showed that the unusual behavior is an artifact of low precision estimates of v ∞ and a long crossover to the scaling law t −1/3 [16].In contrast to the deep understanding of the KPZ class in d = 1 + 1, essentially no exact results are available in d = 2 + 1, the most important dimension for applications [...

We present a numerical study of the evolution of height distributions (HDs) obtained in interface growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. The growth is done on an initially flat substrate. The HDs obtained for all investigated models are very well fitted by the theoretically predicted Gaussian orthogonal ensemble (GOE) distribution. The first cumulant has a shift that vanishes as t(-1/3), while the cumulants of order 2≤n≤4 converge to GOE as t(-2/3) or faster, behaviors previously observed in other KPZ systems. These results yield evidences for the universality of the GOE distribution in KPZ growth on flat substrates. Finally, we further show that the surfaces are described by the Airy(1) process.

The self-affinity of growing systems with radial symmetry, from tumors to grain-grain displacement, has devoted increasing interest in the last decade. In this work, we analyzed features about the interface scaling of these clusters through large scale simulations (up to 3 × 10 7 particles) of two-dimensional growth processes with special emphasis on the off-lattice Eden model. The central objective is to discuss an important pitfall associated to the evaluation of the growth exponent β of these systems. We show that the β value depends on the choice of the origin used to determine the interface width. We considered two strategies frequently used. When the width is evaluated in relation to the center of mass (CM) of the border, the exponent obtained for the Eden model was β CM = 0.404 ± 0.013, in very good agreement with previous reported values. However, if the border CM is replaced by the initial seed position (a static origin), the exponent β 0 = 0.333 ± 0.010, in complete agreement with the KPZ value β KP Z = 1/3, was found. The difference between β CM and β 0 was explained through the border CM fluctuations that grow faster than the overall interface fluctuations. Indeed, we show that the exponents β 0 and β CM characterize large and small wavelength fluctuations of the interface, respectively. These finds were also observed in three distinct lattice models, in which the lattice-imposed anisotropy is absent.

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.