Universal fluctuations in Kardar-Parisi-Zhang growth on one-dimensional flat substrates

Universal interface fluctuations in the contact process

We study the interface representation of the contact process at its directed-percolation critical point, where the scaling properties of the interface can be related to those of the original particle model. Interestingly, such a behavior happens to be intrinsically anomalous and more complex than that described by the standard Family-Vicsek dynamic scaling Ansatz of surface kinetic roughening. We expand on a previous numerical study by Dickman and Muñoz [Phys. Rev. E 62, 7632 (2000)10.1103/PhysRevE.62.7632] to fully characterize the kinetic roughening universality class for interface dimensions d=1,2, and 3. Beyond obtaining scaling exponent values, we characterize the interface fluctuations via their probability density function (PDF) and covariance, seen to display universal properties which are qualitatively similar to those recently assessed for the Kardar-Parisi-Zhang (KPZ) and other important universality classes of kinetic roughening. Quantitatively, while for d=1 the interface covariance seems to be well described by the KPZ, Airy_{1} covariance, no such agreement occurs in terms of the fluctuation PDF or the scaling exponents.

One-point height fluctuations and two-point correlators of (2+1) cylindrical KPZ systems

While the one-point height distributions (HDs) and two-point covariances of (2+1) Kardar-Parisi-Zhang (KPZ) systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat (2+1) KPZ systems, while the latter resembles the Airy_{2} covariance of circular (1+1) KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance A(t/t_{0}) decays asymptotically as A(x)∼x^{-λ[over ¯]} with an exponent λ[over ¯]=β+d^{*}/z, where d^{*} is the number of interface sides kept fixed during the growth (being d^{*}=1 for the systems analyzed here). Overall, these results complete the picture of the main statistics for the (2+1) KPZ class.

Height distributions in interface growth: The role of the averaging process

Height distributions (HDs) are key quantities to uncover universality and geometry-dependence in evolving interfaces. To quantitatively characterize HDs, one uses adimensional ratios of their first central moments (m_{n}) or cumulants (κ_{n}), especially the skewness S and kurtosis K, whose accurate estimate demands an averaging over all L^{d} points of the height profile at a given time, in translation-invariant interfaces, and over N independent samples. One way of doing this is by calculating m_{n}(t) [or κ_{n}(t)] for each sample and then carrying out an average of them for the N interfaces, with S and K being calculated only at the end. Another approach consists in directly calculating the ratios for each interface and, then, averaging the N values. It turns out, however, that S and K for the growth regime HDs display strong finite-size and -time effects when estimated from these "interface statistics," as already observed in some previous works and clearly shown here, through extensive simulations of several discrete growth models belonging to the EW and KPZ classes on one- and two-dimensional substrates of sizes L=const. and L∼t. Importantly, I demonstrate that with "1-point statistics," i.e., by calculating m_{n}(t) [or κ_{n}(t)] once for all NL^{d} heights together, these corrections become very weak, so that S and K attain values very close to the asymptotic ones already at short times and for small L's. However, I find that this "1-point" (1-pt) approach fails in uncovering the universality of the HDs in the steady-state regime (SSR) of systems whose average height, h[over ¯], is a fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt height evolves as h(t)=h[over ¯](t)+s_{λ}A^{1/2}L^{α}ζ+⋯-where P(ζ) is the underlying SSR HD-and the fluctuations in h[over ¯] yield S_{1-pt}∼t^{-1/2} and K_{1-pt}∼t^{-1}. Nonetheless, by analyzing P(h-h[over ¯]), the cumulants of P(ζ) can be accurately determined. I also show that different, but universal, asymptotic values for S and K (related, so, to different HDs) can be found from the "interface statistics" in the SSR. This reveals the importance of employing the various complementary approaches to reliably determine the universality class of a given system through its different HDs.

Spreading fronts of wetting liquid droplets: Microscopic simulations and universal fluctuations

We have used kinetic Monte Carlo (kMC) simulations of a lattice gas to study front fluctuations in the spreading of a nonvolatile liquid droplet onto a solid substrate. Our results are consistent with a diffusive growth law for the radius of the precursor layer, R∼t^{δ}, with δ≈1/2 in all the conditions considered for temperature and substrate wettability, in good agreement with previous studies. The fluctuations of the front exhibit kinetic roughening properties with exponent values which depend on temperature T, but become T independent for sufficiently high T. Moreover, strong evidence of intrinsic anomalous scaling has been found, characterized by different values of the roughness exponent at short and large length scales. Although such a behavior differs from the scaling properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class, the front covariance and the probability distribution function of front fluctuations found in our kMC simulations do display KPZ behavior, agreeing with simulations of a continuum height equation proposed in this context. However, this equation does not feature intrinsic anomalous scaling, at variance with the discrete model.

Universality and crossover behavior of single-step growth models in 1+1 and 2+1 dimensions

We study the kinetic roughening of the single-step (SS) growth model with a tunable parameter p in 1+1 and 2+1 dimensions by performing extensive numerical simulations. We show that there exists a very slow crossover from an intermediate regime dominated by the Edwards-Wilkinson class to an asymptotic regime dominated by the Kardar-Parisi-Zhang (KPZ) class for any p<1/2. We also identify the crossover time, the nonlinear coupling constant, and some nonuniversal parameters in the KPZ equation as a function p. The effective nonuniversal parameters are continuously decreasing with p but not in a linear fashion. Our results provide complete and conclusive evidence that the SS model for p≠1/2 belongs to the KPZ universality class in 2+1 dimensions.

Geometry dependence in linear interface growth

The effect of geometry in the statistics of nonlinear universality classes for interface growth has been widely investigated in recent years, and it is well known to yield a split of them into subclasses. In this work, we investigate this for the linear classes of Edwards-Wilkinson and of Mullins-Herring in one and two dimensions. From comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes, as well as numerical integrations of the growth equations on substrates of fixed size (flat geometry) or expanding linearly in time (radial geometry), we verify that the height distributions (HDs) and the spatial and the temporal covariances are universal but geometry-dependent. In fact, the HDs are always Gaussian, and, when defined in terms of the so-called "KPZ ansatz" [h≃v_{∞}t+(Γt)^{β}χ], their probability density functions P(χ) have mean null, so that all their cumulants are null, except by their variances, which assume different values in the flat and radial cases. The shape of the (rescaled) covariance curves is analyzed in detail and compared with some existing analytical results for them. Overall, these results demonstrate that the splitting of such university classes is quite general, being not restricted to the nonlinear ones.

Circular Kardar-Parisi-Zhang interfaces evolving out of the plane

Circular KPZ interfaces spreading radially in the plane have Gaussian unitary ensemble (GUE) Tracy-Widom (TW) height distribution (HD) and Airy_{2} spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as 〈L(t)〉=L_{0}+ωt^{γ}, while their mean height 〈h〉 increases as usual [〈h〉∼t]. We show that the competition between the L enlargement and the correlation length (ξ≃ct^{1/z}) plays a key role in the asymptotic statistics of the interfaces. While systems with γ>1/z have HDs given by GUE and the interface width increasing as w∼t^{β}, for γ<1/z the HDs are Gaussian, in a correlated regime where w∼t^{αγ}. For the special case γ=1/z, a continuous class of distributions exists, which interpolate between Gaussian (for small ω/c) and GUE (for ω/c≫1). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for ω/c≈10. Despite the GUE HDs for γ>1/z, the spatial covariances present a strong dependence on the parameters ω and γ, agreeing with Airy_{2} only for ω≫1, for a given γ, or when γ=1, for a fixed ω. These results considerably generalize our knowledge on 1D KPZ systems, unveiling the importance of the background space on their statistics.

Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates

Recent experimental works on one-dimensional (1D) circular Kardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reported controversial conclusions about the statistics of their interfaces. Motivated by this, here we investigate several one-dimensional KPZ models on substrates whose size changes in time as L(t)=L_{0}+ωt, focusing on the case ω<0. From extensive numerical simulations, we show that for L_{0}≫1 there exists a transient regime in which the statistics is consistent with that of flat KPZ systems (the ω=0 case), for both ω<0 and ω>0. Actually, for a given model, L_{0} and |ω|, we observe that a difference between ingrowing (ω<0) and outgrowing (ω>0) systems arises only at long times (t∼t_{c}=L_{0}/|ω|), when the expanding surfaces cross over to the statistics of curved KPZ systems, whereas the shrinking ones become completely correlated. A generalization of the Family-Vicsek scaling for the roughness of ingrowing interfaces is presented. Our results demonstrate that a transient flat statistics is a general feature of systems starting with large initial sizes, regardless of their curvature. This is consistent with their recent observation in ingrowing turbulent liquid crystal interfaces, but it is in contrast with the apparent observation of curved statistics in colloidal deposition at the edge of evaporating drops. A possible explanation for this last result, as a consequence of the very small number of monolayers analyzed in this experiment, is given. This is illustrated in a competitive growth model presenting a few-monolayer transient and an asymptotic behavior consistent, respectively, with the curved and flat statistics.

Radial restricted solid-on-solid and etching interface-growth models

An approach to generate radial interfaces is presented. A radial network recursively obtained is used to implement discrete model rules designed originally for the investigation in flat substrates. I used the restricted solid-on-solid and etching models as to test the proposed scheme. The results indicate the Kardar, Parisi, and Zhang conjecture is completely verified leading to a good agreement between the interface radius fluctuation distribution and the Gaussian unitary ensemble. The evolution of the radius agrees well with the generalized conjecture, and the two-point correlation function exhibits also a good agreement with the covariance of the Airy_{2} process. The approach can be used to investigate radial interfaces evolution for many other classes of universality.

Initial pseudo-steady state & asymptotic KPZ universality in semiconductor on polymer deposition

The Kardar-Parisi-Zhang (KPZ) class is a paradigmatic example of universality in nonequilibrium phenomena, but clear experimental evidences of asymptotic 2D-KPZ statistics are still very rare, and far less understanding stems from its short-time behavior. We tackle such issues by analyzing surface fluctuations of CdTe films deposited on polymeric substrates, based on a huge spatio-temporal surface sampling acquired through atomic force microscopy. A pseudo-steady state (where average surface roughness and spatial correlations stay constant in time) is observed at initial times, persisting up to deposition of ~10⁴ monolayers. This state results from a fine balance between roughening and smoothening, as supported by a phenomenological growth model. KPZ statistics arises at long times, thoroughly verified by universal exponents, spatial covariance and several distributions. Recent theoretical generalizations of the Family-Vicsek scaling and the emergence of log-normal distributions during interface growth are experimentally confirmed. These results confirm that high vacuum vapor deposition of CdTe constitutes a genuine 2D-KPZ system, and expand our knowledge about possible substrate-induced short-time behaviors.

Width and extremal height distributions of fluctuating interfaces with window boundary conditions

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size l, for interfaces in several universality classes, in substrate dimensions d_{s}=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when ξ≪l (ξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (ξ/l)^{(n-1)d_{s}}. This gives rise to an interesting temporal scaling for such cumulants as 〈w_{n}〉_{c}∼t^{γ_{n}}, with γ_{n}=2nβ+(n-1)d_{s}/z=[2n+(n-1)d_{s}/α]β. This scaling is analytically proved for the Edwards-Wilkinson (EW) and random deposition interfaces and numerically confirmed for other classes. In general, it is featured by small corrections, and, thus, it yields exponents γ_{n} (and, consequently, α,β and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness α exponents. The stationary (for ξ≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.

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 ɛ 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.

Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension

We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions is obeyed by the restricted solid-on-solid model for substrates with dimensions up to d=6. Analyzing different restriction conditions, we show that the 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.

Relaxation after a change in the interface growth dynamics

The global effects of sudden changes in the interface growth dynamics are studied using models of the Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) classes during their growth regimes in dimensions d=1 and d=2. Scaling arguments and simulation results are combined to predict the relaxation of the difference in the roughness of the perturbed and the unperturbed interfaces, ΔW^{2}∼s{c}t{-γ}, where s is the time of the change and t>s is the observation time after that event. The previous analytical solution for the EW-EW changes is reviewed and numerically discussed in the context of lattice models, with possible decays with γ=3/2 and γ=1/2. Assuming the dominant contribution to ΔW{2} to be predicted from a time shift in the final growth dynamics, the scaling of KPZ-KPZ changes with γ=1-2β and c=2β is predicted, where β is the growth exponent. Good agreement with simulation results in d=1 and d=2 is observed. A relation with the relaxation of a local autoresponse function in d=1 cannot be discarded, but very different exponents are shown in d=2. We also consider changes between different dynamics, with the KPZ-EW as a special case in which a faster growth, with dynamical exponent z_{i}, changes to a slower one, with exponent z. A scaling approach predicts a crossover time t_{c}∼s{z/z_{i}}≫s and ΔW{2}∼s{c}F(t/t_{c}), with the decay exponent γ=1/2 of the EW class. This rules out the simplified time shift hypothesis in d=2 dimensions. These results help to understand the remarkable differences in EW smoothing of correlated and uncorrelated surfaces, and the approach may be extended to sudden changes between other growth dynamics.

Circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer

We consider the Kardar-Parisi-Zhang equation for a circular interface in two dimensions, unconstrained by the standard small-slope and no-overhang approximations. Numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short-time regime with the interface fluctuations of a self-avoiding ring or two-dimensional vesicle, and a long-time regime governed by the Tracy-Widom distribution expected for this geometry. For small-noise amplitudes, scaling behavior is only of the latter type. Large noise is also seen to renormalize the bare physical parameters of the ring, akin to analogous parameter renormalization for equilibrium three-dimensional membranes. Our results bear particular importance on the relation between relevant universality classes of scale-invariant systems in two dimensions.

Universal aspects of curved, flat, and stationary-state Kardar-Parisi-Zhang statistics

Motivated by the recent exact solution of the stationary-state Kardar-Parisi-Zhang (KPZ) statistics by Imamura and Sasamoto [ Phys. Rev. Lett. 108 190603 (2012)], as well as a precursor experimental signature unearthed by Takeuchi [ Phys. Rev. Lett. 110 210604 (2013)], we establish here the universality of these phenomena, examining scaling behaviors of directed polymers in a random medium, the stochastic heat equation with multiplicative noise, and kinetically roughened KPZ growth models. We emphasize the value of cross KPZ-class universalities, revealing crossover effects of experimental relevance. Finally, we illustrate the great utility of KPZ scaling theory by an optimized numerical analysis of the Ulam problem of random permutations.

Interface dynamics of immiscible two-phase lattice-gas cellular automata: a model with random dynamic scatterers and quenched disorder in two dimensions

We use a lattice gas cellular automata model in the presence of random dynamic scattering sites and quenched disorder in the two-phase immiscible model with the aim of producing an interface dynamics similar to that observed in Hele-Shaw cells. The dynamics of the interface is studied as one fluid displaces the other in a clean lattice and in a lattice with quenched disorder. For the clean system, if the fluid with a lower viscosity displaces the other, we show that the model exhibits the Saffman-Taylor instability phenomenon, whose features are in very good agreement with those observed in real (viscous) fluids. In the system with quenched disorder, we obtain estimates for the growth and roughening exponents of the interface width in two cases: viscosity-matched fluids and the case of unstable interface. The first case is shown to be in the same universality class of the random deposition model with surface relaxation. Moreover, while the early-time dynamics of the interface behaves similarly, viscous fingers develop in the second case with the subsequent production of bubbles in the context of a complex dynamics. We also identify the Hurst exponent of the subdiffusive fractional Brownian motion associated with the interface, from which we derive its fractal dimension and the universality classes related to a percolation process.

Crossover from growing to stationary interfaces in the Kardar-Parisi-Zhang class

This Letter reports on how the interfaces in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) class undergo, in the course of time, a transition from the flat, growing regime to the stationary one. Simulations of the polynuclear growth model and experiments on turbulent liquid crystal reveal universal functions of the KPZ class governing this transition, which connect the distribution and correlation functions for the growing and stationary regimes. This in particular shows how interfaces realized in experiments and simulations actually approach the stationary regime, which is never attained unless a stationary interface is artificially given as an initial condition.

Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections

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 one dimension, 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 the KPZ class in two-dimensional systems.

Distribution of scaled height in one-dimensional competitive growth profiles

This work investigates the scaled height distribution, ρ(q), of irregular profiles that are grown based on two sets of local rules: those of the restricted solid on solid (RSOS) and ballistic deposition (BD) models. At each time step, these rules are respectively chosen with probability p and r=1-p. Large-scale Monte Carlo simulations indicate that the system behaves differently in three succeeding intervals of values of p: I(B) ≈ [0,0.75),I(T) ≈ (0.75,0.9), and I(R) ≈ (0.9,1.0]. In I(B), the ballistic character prevails: the growth velocity υ(∞) decreases with p in a linear way, and similar behavior is found for Γ(∞) (p), the amplitude of the t(1/3)-fluctuations, which is measured from the second-order height cumulant. The distribution of scaled height fluctuations follows the Gaussian orthogonal ensemble (GOE) Tracy-Widom (TW) distribution with resolution roughly close to 10(-4). The skewness and kurtosis of the computed distribution coincide with those for TW distribution. Similar results are observed in the interval I(R), with prevalent RSOS features. In this case, the skewness become negative. In the transition interval I(T), the system goes smoothly from one regime to the other: the height distribution becomes apparently Gaussian, which motivates us to identify this phenomenon as a transition from Kardar-Parisi-Zhang (KPZ) behavior to Edwards-Wilkinson (EW) behavior back to KPZ behavior.