In this work we extend the etching model to $d+1$ dimensions. This permits us to investigate its exponents behaviour on higher dimensions, to try to verify the existence of an upper critical dimension for the KPZ equations, with our results sugesting that $d=4$ is not an upper critical dimension for the etching model.
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 Airy$_2$ process. Our simulation results fill the last lacking gap of the conjecture that had been recently verified an…
We report on the first exact solution of the KPZ equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile. The solution provides a determinantal formula for the probability distribution function of the height $h(x,t)$ for all $t>0$. In particular, we show that for large $t$, on the scale $t^{1/3}$, the statistics is given by the Tracy-Widom distribution, known already from the theory of GUE random matrices. Our solution confirms that the KPZ equation describes the interface motion in the regime of weak driving fo…
Scanning probe microscopy (SPM) is a fundamental technique for the analysis of surfaces. In the present work, the interface statistics of surfaces scanned with a probe tip was analyzed for both \textit{in silico} and experimental systems that \textit{do not} belong to the prominent Kardar-Parisi-Zhang (KPZ) universality class. We show that height, local roughness and extremal height distributions of scanned surfaces quantitatively agree with the KPZ class in a range similar or better than recent experimental evidences of the KPZ class using SPM images. The underlying mechanism behind this arti…
Self-sustained reaction fronts in a disordered medium subject to an external flow display self-affine roughening, pinning and depinning transitions. We measure spatial and temporal fluctuations of the front in $1+1$ dimensions, controlled by a single parameter, the mean flow velocity. Three distinct universality classes are observed, consistent with the Kardar-Parisi-Zhang (KPZ) class for fast advancing or receding fronts, the quenched KPZ class (positive-qKPZ) when the mean flow approximately cancels the reaction rate, and the negative-qKPZ class for slowly receding fronts. Both quenched KPZ …
We introduce the notion of energy solutions of the KPZ equation. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, weakly asymmetric, conservative particle systems with respect to the stationary states are given by energy solutions of the KPZ equation. As a consequence, we prove that the Cole-Hofp solutions are also energy solutions of the KPZ equation.
We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $γ$ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a non-trivial effect on the probability distribution of the random walker. At small $γ$, we show analytically that the time-evolution of the latter can be mapped to the Stochastic Heat Equation (SHE). In this limit, the width of the log probability thus follows a Family-Vicsek scaling law, $N^αf(t/N^{α/β})$, with roughness and growth exponents corresponding to the K…
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 $2\le n\le 4$ converge to GOE as $t^{-2/3}$ or faster, behaviors previously observed in other KPZ systems. These results yield a new eviden…
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 $p <\frac{1}{2}$. 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 w…
Simulations of restricted solid-on-solid growth models are used to build the width-distributions of d=2-5 dimensional KPZ interfaces. We find that the universal scaling function associated with the steady-state width-distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity.
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 \textit{pseudo}-steady state (where average surface roughness and spatial correlations stay constant in time) is observed at initial times, persis…
We present large-scale simulations of radial Eden clusters in three-dimensions and show that the growth exponent is in agreement with the value $β=0.242$ accepted for the Kardar-Parisi-Zhang (KPZ) universality class. Our results refute a recent assertion proposing that radial Eden growth in $d=3$ belongs to a universality class distinct from KPZ. We associate the previously reported discrepancy to a slow convergence to the asymptotic limit. We also present the skewness and kurtosis in the roughening regime for flat geometry in 2+1 dimensions.
We revisit the anchored Toom interface and use KPZ scaling theory to argue that the interface fluctuations are governed by the Airy_1 process with the role of space and time interchanged. There is no free parameter. The predictions are numerically well confirmed for space-time statistics in the stationary state. In particular the spatial fluctuations of the interface are given by the GOE edge distribution of Tracy and Widom.
We report on the growth dynamic of CdTe thin films for deposition temperatures ($T$) in the range of $150\,^{\circ}\mathrm{C}$ to $300\,^{\circ}\mathrm{C}$. A relation between mound evolution and large-wavelength fluctuations at CdTe surface has been established. One finds that short-length scales are dictated by an interplay between the effects of the formation of defects at boundaries of neighbouring grains and a relaxation process which stems from the diffusion and deposition of particles (CdTe molecules) torward these regions. A Kinetic Monte Carlo model corroborates these reasonings. As $…
We present a lattice gas model that without fine tuning of parameters is expected to exhibit the so far elusive modified Kardar-Parisi-Zhang (KPZ) universality class. To this end, we review briefly how non-linear fluctuating hydrodynamics in one dimension predicts that all dynamical universality classes in its range of applicability belong to an infinite discrete family which we call Fibonacci family since their dynamical exponents are the Kepler ratios $z_i = F_{i+1}/F_{i}$ of neighbouring Fibonacci numbers $F_i$, including diffusion ($z_2=2$), KPZ ($z_3=3/2$), and the limiting ratio which is…
We assess the dependence on substrate dimensionality of the asymptotic scaling behavior of a whole family of equations that feature the basic symmetries of the Kardar-Parisi-Zhang (KPZ) equation. Even for cases in which, as expected from universality arguments, these models display KPZ values for the critical exponents and limit distributions, their behavior deviates from KPZ scaling for increasing system dimensions. Such a fragility of KPZ universality contradicts naive expectations, and questions straightforward application of universality principles for the continuum description of experime…
Consider a deterministically growing surface of any dimension, where the growth at a point is an arbitrary nonlinear function of the heights at that point and its neighboring points. Assuming that this nonlinear function is monotone, invariant under the symmetries of the lattice, equivariant under constant shifts, and twice continuously differentiable, it is shown that any such growing surface approaches a solution of the deterministic KPZ equation in a suitable space-time scaling limit.
The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation statistics. Its space derivative, namely the noisy Burgers equation, has played a very important role in its study, predating the formulation of the KPZ equation proper, and being frequently held as an equivalent system. We show that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the fluctuations behave in a remarkably different way: whi…
We report on the universality of height fluctuations at the crossing point of two interacting (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) interfaces with curved and flat initial conditions. We introduce a control parameter p as the probability for the initially flat geometry to be chosen and compute the phase diagram as a function of p. We find that the distribution of the fluctuations converges to the Gaussian orthogonal ensemble Tracy-Widom (TW) distribution for p<0.5, and to the Gaussian unitary ensemble TW distribution for p>0.5. For p=0.5 where the two geometries are equally weighted, the…
Universality is a well-established central concept of equilibrium physics. However, in systems far away from equilibrium a deeper understanding of its underlying principles is still lacking. Up to now, a few classes have been identified. Besides the diffusive universality class with dynamical exponent $z=2$ another prominent example is the superdiffusive Kardar-Parisi-Zhang (KPZ) class with $z=3/2$. It appears e.g. in low-dimensional dynamical phenomena far from thermal equilibrium which exhibit some conservation law. Here we show that both classes are only part of an infinite discrete family …
It has recently been proposed that fluctuating ``pulled'' fronts propagating into an unstable state should not be in the standard KPZ universality class for rough interface growth. We introduce an effective field equation for this class of problems, and show on the basis of it that noisy pulled fronts in {\em d+1} bulk dimensions should be in the universality class of the {\em (d+1)+1}D KPZ equation rather than of the {\em d+1}D KPZ equation. Our scenario ties together a number of heretofore unexplained observations in the literature, and is supported by previous numerical results.
The Kardar-Parisi-Zhang (KPZ) equation defines the main universality class for nonlinear growth and roughening of surfaces. But under certain conditions, a conserved KPZ equation (cKPZ) is thought to set the universality class instead. This has non-mean-field behavior only in spatial dimension $d<2$. We point out here that cKPZ is incomplete: it omits a symmetry-allowed nonlinear gradient term of the same order as the one retained. Adding this term, we find a parameter regime where the $1$-loop renormalization group flow diverges. This suggests a phase transition to a new growth phase, possibl…
These notes are based on lectures delivered by the authors at a Langeoog seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a mixed audience of mathematicians and theoretical physicists. After a brief outline of the basic physical concepts of equilibrium and nonequilibrium states, the one-dimensional simple exclusion process is introduced as a paradigmatic nonequilibrium interacting particle system. The stationary measure on the ring is derived and the idea of the hydrodynamic limit is sketched. We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation a…
Numerical results for the Directed Polymer model in 1+4 dimensions in various types of disorder are presented. The results are obtained for system size considerably larger than that considered previously. For the extreme strong disorder case (Min-Max system), associated with the Directed Percolation model, the expected value of the meandering exponent, zeta = 0.5 is clearly revealed, with very week finite size effects. For the week disorder case, associated with the KPZ equation, finite size effects are stronger, but the value of seta is clearly seen in the vicinity of 0.57. In systems with "s…
On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph
Universidad de Extremadura（埃斯特雷马杜拉大学）；Instituto de Computación Científica Avanzada de Extremadura (ICCAEx), Universidad de Extremadura（埃斯特雷马杜拉先进计算研究所 (ICCAEx)）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
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 $d\ge7$ predicts that the upper critical dimension of the KPZ class is infin…
The power spectrum of interface fluctuations in the $(1+1)$-dimensional Kardar-Parisi-Zhang (KPZ) universality class is studied both experimentally and numerically. The $1/f^α$-type spectrum is found and characterized through a set of "critical exponents" for the power spectrum. The recently formulated "aging Wiener-Khinchin theorem" accounts for the observed exponents. Interestingly, the $1/f^α$ spectrum in the KPZ class turns out to contain information on a universal distribution function characterizing the asymptotic state of the KPZ interfaces, namely the Baik-Rains universal variance. It …
We study the restricted solid-on-solid (RSOS) model with finite hopping distance $l_{0}$, using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E {\bf 62}, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the $l_{0}=\infty$ case which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of $l_{0}$. Consequently, we find that the CRSOS model belongs to the VLD universality class and the RSOS models with any finite hopping distance be…
The front of a compact bacterial colony growing on a Petri dish is a paradigmatic instance of non-equilibrium fluctuations in the celebrated Eden, or Kardar-Parisi-Zhang (KPZ), universality class. While in many experiments the scaling exponents crucially differ from the expected KPZ values, the source of this disagreement has remained poorly understood. We have performed growth experiments with B. subtilis 168 and E. coli ATCC 25922 under conditions leading to compact colonies in the classically-alleged Eden regime, where individual motility is suppressed. Non-KPZ scaling is indeed observed fo…
Aiming to investigate the upper critical dimension, $d_u$, of the KPZ class, in [EPL 103 (2013) 10005] some growth models were numerically analyzed using Cayley trees (CTs) as substrates, as a way to access their behavior in the infinite-dimensional limit, and some unexpected results were reported: logarithmic roughness scaling, differing for EW and KPZ models (indicating that even at $d=\infty$ the KPZ nonlinearity is still relevant); beyond asymptotically rough EW surfaces above the upper critical dimension of the EW class. Motivated by these strange findings, I revisit these growth models h…
Discrete and continuous models belonging to a universality class share the same linearities and (or) nonlinearities. In this work, we propose a new approach to calculate coarse grained coefficients of the continuous differential equation from discrete models. We apply small constant translations in a test space and show how to obtain these coefficients from the transformed average interface growth velocity. Using the examples of the ballistic deposition (BD) model and the restricted solid-on-solid (RSOS) model, both belonging to the Kardar-Parisi-Zhang (KPZ) universality class, we demonstrate …
The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades. Some experiments indicated the power law for the interface width, $w(t)\sim t^β$, remains the same as in growth on planar surfaces. Escudero (Phys. Rev. Lett. {\bf 100}, 116101, 2008) argued, however, that for the radial KPZ equations in (1+1)-dimension $w(t)$ should increase as $w(t)\sim [\ln(t)]^{1/2}$ in the long-time limit. Krug (Phys. Rev. Lett. {\bf 102}, 139601, 2009) argued, however, that the dynamics of the interface must remain unchanged with a change in the geometry. Other studies ind…
The statistics of the fluctuations of quantum many-body systems are highly revealing of their nature. In driven-dissipative systems displaying macroscopic quantum coherence, as exciton polariton condensates under incoherent pumping, the phase dynamics can be mapped to the stochastic Kardar-Parisi-Zhang (KPZ) equation. However, in two dimensions (2D), it was theoretically argued that the KPZ regime may be hindered by the presence of vortices, and a non-equilibrium BKT behavior was reported close to condensation threshold. We demonstrate here that, when a discretized 2D polariton system is consi…
Three-Dimensional Kardar--Parisi--Zhang Scaling in Polariton Condensates
Moscow Center for Advanced Studies（莫斯科高级研究中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；Barnard College, Columbia University（哥伦比亚大学巴纳德学院）；Westlake University（西湖大学）；St. Petersburg State University（圣彼得堡国立大学）；Russian Quantum Center（俄罗斯量子中心）
The one dimensional Kardar-Parisi-Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a natural renormalization/rescaling on the space of such evolving interfaces. We introduce and describe the renormalization fixed point of the Kardar-Parisi-Zhang universality class in terms of a random nonlinear semigroup with stationary independent increments, and via a variational formula. Furthermore, we compute a plausible formula the exact transition probabilities using replica Bethe ansatz. The …
The shape of large on-lattice Eden clusters grown from a single seed is ruled by the underlying lattice anisotropy. This is reflected on the linear growth with time of the interface width ($w\sim t$), in contrast with the KPZ universality class ($w\sim t^{1/3}$) observed when the Eden model is grown on flat substrates. We propose an extended Eden model, in which the growth probability has a power law dependence with the number of occupied nearest neighbors. Large scale simulations ($N\gtrsim 4\times 10^9$ particles) were used to determine the time evolution of $w$. We found that a suitable cho…
Revealing universal behaviors is a hallmark of statistical physics. Phenomena such as the stochastic growth of crystalline surfaces, of interfaces in bacterial colonies, and spin transport in quantum magnets all belong to the same universality class, despite the great plurality of physical mechanisms they involve at the microscopic level. This universality stems from a common underlying effective dynamics governed by the non-linear stochastic Kardar-Parisi-Zhang (KPZ) equation. Recent theoretical works suggest that this dynamics also emerges in the phase of out-of-equilibrium systems displayin…
The determination of the exact exponents of the KPZ class in any substrate dimension $d$ is one of the most important open issues in Statistical Physics. Based on the behavior of the dimensional variation of some exact exponent differences for other growth equations, I find here that the KPZ growth exponents (related to the temporal scaling of the fluctuations) are given by $β_d = \frac{7}{8d+13}$. These exponents present an excellent agreement with the most accurate estimates for them in the literature. Moreover, they are confirmed here through extensive Monte Carlo simulations of discrete gr…
We study the role of the topology of the background space on the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class. To do so, we study the growth of balls on disordered 2D manifolds with random Riemannian metrics, generated by introducing random perturbations to a base manifold. As base manifolds we consider cones of different aperture angles $θ$, including the limiting cases of a cylinder ($θ=0$, which corresponds to an interface with periodic boundary conditions) and a plane ($θ=π/2$, which corresponds to an interface with circular geometry). We obtain that in the former case the …
We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, on flat (fixed-size) and expanding substrates (ES). In both $d=1+1$ and $2+1$, we find that growth regime height distributions (HDs), and spatial and temporal covariances are universal, but are dependent on the initial conditions, while the critical exponents are the same for flat and ES systems. Thus, the nMBE class does split into subclasses, as does the Kardar-Parisi-Zhang (KPZ) class. Applying the "KPZ ansatz" to nMBE models, we estimate the cumulants of the $1+1$ HDs.…
We investigate the statistics of Iso-height lines of (2+1)-dimensional Kardar-Parisi-Zhang model at different level sets around the mean height in the saturation regime. We find that the exponent describing the distribution of the height-cluster size behaves differently for level cuts above and below the mean height, while the fractal dimensions of the height-clusters and their perimeters remain unchanged. The winding angle statistics also confirms again the conformal invariance of these contour lines in the same universality class of self-avoiding random walks (SAWs).
Dynamical universality classes are distinguished by their dynamical exponent $z$ and unique scaling functions encoding space-time asymmetry for, e.g. slow-relaxation modes or the distribution of time-integrated currents. So far the universality class of the Nagel-Schreckenberg (NaSch) model, which is a paradigmatic model for traffic flow on highways, was not known except for the special case $v_{\text{max}}=1$. Here the model corresponds to the TASEP (totally asymmetric simple exclusion process) that is known to belong to the superdiffusive Kardar-Parisi-Zhang (KPZ) class with $z=3/2$. In this…
The recently developed formalism of nonlinear fluctuating hydrodynamics (NLFH) has been instrumental in unraveling many new dynamical universality classes in coupled driven systems with multiple conserved quantities. In principle, this formalism requires knowledge of the exact expression of locally conserved current in terms of local density of the conserved components. However, for most nonequilibrium systems an exact expression is not available and it is important to know what happens to the predictions of NLFH in these cases. We address this question for the first time here in a system with…
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 βis the growth exponent. Our results support a generalization of the ansatz h = v t + (Γt)^βχ+ η+ ζt^-βto higher dimensions, where v, Γ, ζand ηare non-universal quantities whereas βand χare un…
We consider the evolution of interfaces with a diffusive term and a generalized Kardar-Parisi-Zhang (KPZ) non-linearity, which results in a propagation velocity that depends periodically on the tilt of the interface. Using large scale simulations of a model class with these properties in 1+1 dimensions, we show that the fluctuations are in general still in the KPZ universality class, but a new universality class seems to appear in the limit of weak non-linearity. We argue that this is the typical behavior of any interface model with periodic tilt dependence.
Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) \textit{stochastic} partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state (NESS) of the one-dimensional Kuramoto-Sivashinsky (KS) \textit{deterministic} PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile $h(x,t)$ for different initial conditions. We es…
Starting from a Langevin formulation of a thermally perturbed nonlinear elastic model of the ferroelectric smectic-C$^*$ (SmC${*}$) liquid crystals in the presence of an electric field, this article characterizes the hitherto unexplored dynamical phase transition from a thermo-electrically forced ferroelectric SmC${}^{*}$ phase to a chiral nematic liquid crystalline phase and vice versa. The theoretical analysis is based on a combination of dynamic renormalization (DRG) and numerical simulation of the emergent model. While the DRG architecture predicts a generic transition to the Kardar-Parisi…
Exciton-polaritons under driven-dissipative conditions exhibit a condensation transition which belongs to a different universality class than equilibrium Bose-Einstein condensates. By numerically solving the generalized Gross-Pitaevskii equation with realistic experimental parameters, we show that one-dimensional exciton-polaritons display fine features of Kardar-Parisi-Zhang (KPZ) dynamics. Beyond the scaling exponents, we show that their phase distribution follows the Tracy-Widom form predicted for KPZ growing interfaces. We moreover evidence a crossover to the stationary Baik-Rains statisti…
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class …
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. The ansatz h = v t + (Γt)^{1/3} χ+ η, where χis a Tracy-Widom (geo…
We study KPZ surfaces on Euclidean lattices and directed polymers on hierarchical lattices subject to different distributions of disorder, showing that universality holds, at odds with recent results on Euclidean lattices. Moreover, we find the presence of a slow (power-law) crossover toward the universal values of the exponents and verify that the exponent governing such crossover is universal too. In the limit of a 1+epsilon dimensional system we obtain both numerically and analytically that the crossover exponent is 1/2.
We introduce a self-organized surface growth model in 2+1 dimensions with anisotropic avalanche process, which is expected to be in the universality class of the anisotropic quenched Kardar-Parisi-Zhang equation with alternative signs of the nonlinear KPZ terms. It turns out that the surface height correlation functions in each direction scales distinctively. The anisotropic behavior is attributed to the asymmetric behavior of the quenched KPZ equation in 1+1 dimensions with respect to the sign of the nonlinear KPZ term.
In order to estimate roughness exponents of interface growth models, we propose the calculation of effective exponents from the roughness fluctuation (sigma) in the steady state. We compare the finite-size behavior of these exponents and the ones calculated from the average roughness <w_2> for two models in the 2+1-dimensional Kardar-Parisi-Zhang (KPZ) class and for a model in the 1+1-dimensional Villain-Lai-Das Sarma (VLDS) class. The values obtained from sigma provide consistent asymptotic estimates, eventually with smaller finite-size corrections. For the VLDS (nonlinear molecular beam epit…
