We show that a 2+1 dimensional discrete surface growth model exhibiting Kardar-Parisi-Zhang (KPZ) class scaling can be mapped onto a two dimensional conserved lattice gas model of directed dimers. In case of KPZ height anisotropy the dimers follow driven diffusive motion. We confirm by numerical simulations that the scaling exponents of the dimer model are in agreement with those of the 2+1 dimensional KPZ class. This opens up the possibility of analyzing growth models via reaction-diffusion models, which allow much more efficient computer simulations.
Extended dynamical simulations have been performed on a 2+1 dimensional driven dimer lattice gas model to estimate ageing properties. The auto-correlation and the auto-response functions are determined and the corresponding scaling exponents are tabulated. Since this model can be mapped onto the 2+1 dimensional Kardar-Parisi-Zhang surface growth model, our results contribute to the understanding of the universality class of that basic system.
We propose a unified moving boundary problem for surface growth by electrochemical and chemical vapor deposition, which is derived from constitutive equations into which stochastic forces are incorporated. We compute the coefficients in the interface equation of motion as functions of phenomenological parameters. The equation features the Kardar-Parisi-Zhang (KPZ) non-linearity and instabilities which, depending on surface kinetics, can hinder the asymptotic KPZ scaling. Our results account for the universality and the experimental scarcity of KPZ scaling in the growth processes considered.
We present results from extensive numerical integration of the KPZ equation in $1 + 1$ dimensions aimed to check the long-time behavior of the dynamical structure factor of that system. Over a number of decades in the size of the structure factor we confirm scaling and stretched exponential decay. We also give an analytic expression that yields a very good approximation to the numerical data.
Local Scale-Invariance theory is tested by extensive dynamical simulations of the driven dimer lattice gas model, describing the surface growth of the 2+1 dimensional Kardar-Parisi-Zhang surfaces. Very precise measurements of the universal autoresponse function enabled us to perform nonlinear fitting with the scaling forms, suggested by local scale-invariance (LSI). While the simple LSI ansatz does not seem to work, forms based on logarithmic extension of LSI provide satisfactory description of the full (measured) time evolution of the autoresponse function.
We review a recent asymptotic weak noise approach to the Kardar-Parisi-Zhang equation for the kinetic growth of an interface in higher dimensions. The weak noise approach provides a many body picture of a growing interface in terms of a network of localized growth modes. Scaling in 1d is associated with a gapless domain wall mode. The method also provides an independent argument for the existence of an upper critical dimension.
We apply a numerical minimum action method derived from the Wentzell-Freidlin theory of large deviations to the Kardar-Parisi-Zhang equation for a growing interface. In one dimension we find that the switching scenario is determined by the nucleation and subsequent propagation of facets or steps, corresponding to moving domain walls or growth modes in the underlying noise driven Burgers equation. The transition scenario is in accordance with recent analytical studies of the one dimensional Kardar-Parisi-Zhang equation in the asymptotic weak noise limit. We also briefly discuss transitions in t…
The controversy whether or not he Kardar-Parisi-Zhang (KPZ) equation has an upper critical dimension (UCD) is going on for quite a long time. Some approximate integral equations for the two-point function served as an indication for the existence of a UCD, by obtaining a dimension, above which the equation does not have a strong coupling solution. A surprising aspect of these studies, however, is that variuos authors that considered the same equation produced large variations in the UCD. This caused some doubts concerning the existence of a UCD. Here we revisit these calculations, describe the…
We investigate analytically the large dimensional behavior of the Kardar-Parisi-Zhang (KPZ) dynamics of surface growth using a recently proposed non-perturbative renormalization for self-affine surface dynamics. Within this framework, we show that the roughness exponent $α$ decays not faster than $α\sim 1/d$ for large $d$. This implies the absence of a finite upper critical dimension.
The Kardar-Parisi-Zhang (KPZ) equation has been connected to a large number of important stochastic processes in physics, chemistry and growth phenomena, ranging from classical to quantum physics. The central quest in this field is the search for ever more precise universal growth exponents. Notably, exact growth exponents are only known for $1+1$ dimensions. In this work, we present physical and geometric analytical methods that directly associate these exponents to the fractal dimension of the rough interface. Based on this, we determine the growth exponents for the $2+1$ dimensions, which a…
In this paper various predictions for the scaling exponents of the Nonlocal Kardar-Parisi-Zhang (NKPZ) equation are discussed. I use the Self-Consistent Expansion (SCE), and obtain results that are quite different from result obtained in the past, using Dynamic Renormalization Group analysis (DRG), a Scaling Approach (SA) and a self-consistent Mode Coupling approach (MC). It is shown that the results obtained using SCE recover an exact result for a subfamily of the NKPZ models in one dimension, while all the other methods fail to do so. It is also shown that the SCE result is the only one that…
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…
We study unbinding transitions of a non-equilibrium Kardar-Parisi-Zhang (KPZ) interface in the presence of long-ranged substrates. Both attractive and repulsive substrates, as well as positive and negative Kardar-Parisi-Zhang nonlinearities, are considered, leading to four different physical situations. A detailed comparison with equilibrium wetting transitions as well as with non-equilibrium unbinding transitions in systems with short-ranged forces is presented, yielding a comprehensive picture of unbinding transitions, and of their classification into universality classes. These non-equilibr…
The article covers the one-dimensional Kardar-Parisi-Zhang equation, weak drive limit, universality, directed polymers in a random medium, replica solutions, statistical mechanics of line ensembles, and its generalization to several components which is used to study equilibrium time correlations of anharmonic chains and of the discrete nonlinear Schrödinger equation. The notes are an extended version of my lectures at Les Houches July 2015.
Ballistic deposition (BD) is considered to be a paradigmatic discrete growth model that represents the Kardar-Parisi-Zhang (KPZ) universality class. In this paper we question this connection by rigorously deriving a formal continuum equation from the BD microscopic rules, which deviates from the KPZ equation. In one dimension these deviations are not important in the presence of noise, but for higher dimensions or when considering deterministic evolution they are very relevant.
The continuum Kardar-Parisi-Zhang equation in one dimension is lattice discretized in such a way that the drift part is divergence free. This allows to determine explicitly the stationary measures. We map the lattice KPZ equation to a bosonic field theory which has a cubic anti-hermitian nonlinearity. Thereby it is established that the stationary two-point function spreads superdiffusively.
I analyze the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation with spatially correlated noise. This equation is also known as the Nonlocal Molecular Beam Epitaxy (NMBE) equation andv was originally suggested to study the effect of the long-range nature of interactions coupled with spatially correlated noise on the dynamics of a volume conserving surface. I find an exact result for a subfamily of NCKPZ models in one dimension, and thus establish an exact result for MBE processes for the first time. Then, to complete the picture, I construct a Self-Consistent Expansion (SCE) and get resu…
We explain the exact solution of the 1+1 dimensional Kardar-Parisi-Zhang equation with sharp wedge initial conditions. Thereby it is confirmed that the continuum model belongs to the KPZ universality class, not only as regards to scaling exponents but also as regards to the full probability distribution of the height in the long time limit.
We present a simple approximation of the non-perturbative renormalization group designed for the Kardar-Parisi-Zhang equation and show that it yields the correct phase diagram, including the strong-coupling phase with reasonable scaling exponent values in physical dimensions. We find indications of a possible qualitative change of behavior around $d=4$. We discuss how our approach can be systematically improved.
We use the $(1+1)$-dimensional Kardar-Parisi-Zhang equation driven by a Gaussian white noise and employ the dynamic renormalization-group of Yakhot and Orszag without rescaling [J.~Sci.\ Comput.~{\bf 1}, 3 (1986)]. Hence we calculate the second and third order moments of height distribution using the diagrammatic method in the large scale and long time limits. The moments so calculated lead to the value $S=0.3237$ for the skewness. This value is comparable with numerical and experimental estimates.
We re-examine mode-coupling theory for the Kardar-Parisi-Zhang (KPZ) equation in the strong coupling limit and show that there exists two branches of solutions. One branch (or universality class) only exists for dimensionalities $d<d_c=2$ and is similar to that found by a variety of analytic approaches, including replica symmetry breaking and Flory-Imry-Ma arguments. The second branch exists up to $d_c=4$ and gives values for the dynamical exponent $z$ similar to those of numerical studies for $d\ge2$.
The emergence of superdiffusive spin dynamics in integrable classical and quantum magnets is well established by now, but there is no generally valid theoretical explanation for this phenomenon. A fundamental difficulty is that the hydrodynamic fluctuations of conserved quasiparticle modes are purely diffusive. We argue that in isotropic integrable magnets, a complete hydrodynamic description must include soft "gauge" degrees of freedom, that arise from spontaneous breaking of the Bethe pseudovacuum symmetry. We show that the coarse-grained time evolution of these modes lies in the Kardar-Pari…
This is the article by the same name which was published in the March 2016 issue of the AMS Notices. Due to space constraints, some references contained here were not provided in the published version were not provided. The figures in this article were made collaboratively with William Casselman.
A nonperturbative weak noise scheme is applied to the Kardar-Parisi-Zhang equation for a growing interface in all dimensions. It is shown that the growth morphology can be interpreted in terms of a dynamically evolving texture of localized growth modes with superimposed diffusive modes. Applying Derrick's theorem it is conjectured that the upper critical dimension is four.
The effects of spatially correlated noise on a phenomenological equation equivalent to a non-local version of the Kardar-Parisi-Zhang equation are studied via the dynamic renormalization group (DRG) techniques. The correlated noise coupled with the long ranged nature of interactions prove the existence of different phases in different regimes, giving rise to a range of roughness exponents defined by their corresponding critical dimensions. Finally self-consistent mode analysis is employed to compare the non-KPZ exponents obtained as a result of the long range -long range interactions with the …
In this paper I study a model for a growing surface in the presence of anomalous diffusion, also known as the Fractal Kardar-Parisi-Zhang equation (FKPZ). This equation includes a fractional Laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a Levy flight. I show that for a specific choice of parameters of the FKPZ equation, the equation can be solved exactly in one dimension, so that all the critical exponents, which describe the surface that grows under FKPZ, can be derived for that case. Afterwards, I use a Self-Consistent Expansion (SCE) …
We introduce a field-theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a Markovian dynamics on a discrete set of states and for overdamped Langevin equations. We first formulate a framework which describes quantities like current, entropy production and diffusivity in the case of a generic field theory. We will then apply this general setting to the one-dimensional Kardar-Parisi-Zhang equation, a paradigmatic example of a non-linear field-theoretic Langevin equation. In particular, we will treat the dimensionless Kardar-Parisi-Zhang equation with an eff…
We obtain the first exact solution for the stationary one-dimensional Kardar-Parisi-Zhang equation. A formula for the distribution of the height is given in terms of a Fredholm determinant, which is valid for any finite time $t$. The expression is explicit and compact enough so that it can be evaluated numerically. Furthermore, by extending the same scheme, we find an exact formula for the stationary two-point correlation function.
The conserved Kardar-Parisi-Zhang equation in the presence of long-range nonlinear interactions is studied by the dynamic renormalization group method. The long-range effect produces new fixed points with continuously varying exponents and gives distinct phase transitions, depending on both the long-range interaction strength and the substrate dimension $d$. The long-range interaction makes the surface width less rough than that of the short-range interaction. In particular, the surface becomes a smooth one with a negative roughness exponent at the physical dimension d=2.
We study the dynamics of vortices in a two-dimensional, non-equilibrium system, described by the compact Kardar-Parisi-Zhang equation, after a sudden quench across the critical region. Our exact numerical solution of the phase-ordering kinetics shows that the unique interplay between non-equilibrium and the variable degree of spatial anisotropy leads to different critical regimes. We provide an analytical expression for the vortex evolution, based on scaling arguments, which is in agreement with the numerical results, and confirms the form of the interaction potential between vortices in this …
We study the noisy nonequilibrium dynamics of a conserved density that is driven by a fluctuating surface governed by the conserved Kardar-Parisi-Zhang equation. We uncover the universal scaling properties of the conserved density. We consider two separate minimal models where the surface fluctuations couple (i) with the spatial variation of the conserved density, and (ii) directly with the magnitude of the conserved density. Both these two models conserve the density, but differ from symmetry stand point. We use our result to highlight the dependence of nonequilibrium universality classes on …
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 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.
We show that d+1-dimensional surface growth models can be mapped onto driven lattice gases of d-mers. The continuous surface growth corresponds to one dimensional drift of d-mers perpendicular to the (d-1)-dimensional "plane" spanned by the d-mers. This facilitates efficient, bit-coded algorithms with generalized Kawasaki dynamics of spins. Our simulations in d=2,3,4,5 dimensions provide scaling exponent estimates on much larger system sizes and simulations times published so far, where the effective growth exponent exhibits an increase. We provide evidence for the agreement with field theoret…
We study the mode-coupling theory for the Kardar-Parisi-Zhang equation in the strong-coupling regime, focusing on the long time properties. By a saddle point analysis of the mode-coupling equations, we derive exact results for the correlation function in the long time limit - a limit which is hard to study using simulations. The correlation function at wavevector k in dimension d is found to behave asymptotically at time t as C(k,t)\simeq 1/k^{d+4-2z} (Btk^z)^{γ/z} e^{-(Btk^z)^{1/z}}, with γ=(d-1)/2, A a determined constant and B a scale factor.
We study the stochastically driven conserved Kardar-Parisi-Zhang (CKPZ) equation with quenched disorders. Short-ranged quenched disorders is found to be a relevant perturbation on the pure CKPZ equation at one dimension, and as a result, a new universality class different from pure CKPZ equation appears to emerge. At higher dimensions, quenched disorder turns out to be ineffective to influence the universal scaling. This results in the asymptotic long wavelength scaling to be given by the linear theory, a scenario identical with the pure CKPZ equation. For sufficiently long-ranged quenched dis…
We study the interface dynamics of a discrete model to quantitatively describe electrochemical deposition experiments. Extensive numerical simulations indicate that the interface dynamics is unstable at early times, but asymptotically displays the scaling of the Kardar-Parisi-Zhang universality class. During the time interval in which the surface is unstable, its power spectrum is anomalous; hence the behaviors at length scales smaller than or comparable with the system size are described by different roughness exponents. These results are expected to apply to a wide range of electrochemical d…
We report on the effect of substrate temperature (T) on both local structure and long-wavelength fluctuations of polycrystalline CdTe thin films deposited on Si(001). A strong T-dependent mound evolution is observed and explained in terms of the energy barrier to inter-grain diffusion at grain boundaries, as corroborated by Monte Carlo simulations. This leads to transitions from uncorrelated growth to a crossover from random-to-correlated growth and transient anomalous scaling as T increases. Due to these finite-time effects, we were not able to determine the universality class of the system t…
The octahedron model introduced recently has been implemented onto graphics cards, which permits extremely large scale simulations via binary lattice gases and bit coded algorithms. We confirm scaling behaviour belonging to the 2d Kardar-Parisi-Zhang universality class and find a surface growth exponent: beta=0.2415(15) on 2^17 x 2^17 systems, ruling out beta=1/4 suggested by field theory. The maximum speed-up with respect to a single CPU is 240. The steady state has been analysed by finite size scaling and a growth exponent alpha=0.393(4) is found. Correction to scaling exponents are computed…
We study scaling properties of the honeycomb fully packed loop ensemble associated with a lozenge tiling model of rough surface, when the latter is driven out of equilibrium by Kardar-Parisi-Zhang (KPZ) type dynamics. We show numerically that conformal invariance and signatures of critical percolation appear in the stationary KPZ state. In terms of the two-component Coulomb gas description of the Edwards-Wilkinson stationary state, our finding is understood as the invariance of one component under the effect of the non-linear KPZ term. On the other hand, we show a breaking of conformal invaria…
We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime we find the strong coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at non-integer dimensions. Apart from the well-known weak coupling and the now well established isotropic strong coupling behavior, we find an anisotropic strong coupling fixed point for nonlinear couplings of opposite signs at non-integer dimensions.
We study numerically the Kuramoto-Sivashinsky (KS) equation forced by external white noise in two space dimensions, that is a generic model for e.g. surface kinetic roughening in the presence of morphological instabilities. Large scale simulations using a pseudospectral numerical scheme allow us to retrieve Kardar-Parisi-Zhang (KPZ) scaling as the asymptotic state of the system, as in the 1D case. However, this is only the case for sufficiently large values of the coupling and/or system size, so that previous conclusions on non-KPZ asymptotics are demonstrated as finite size effects. Crossover…
We introduce the asymmetric extension of the Quantum Symmetric Simple Exclusion Process which is a stochastic model of fermions on a lattice hopping with random amplitudes. In this setting, we analytically show that the time-integrated current of fermions defines a height field which exhibits a quantum non-linear stochastic Kardar-Parisi-Zhang dynamics. Similarly to classical simple exclusion processes, we further introduce the discrete Cole-Hopf (or Gärtner) transform of the height field which satisfies a quantum version of the Stochastic Heat Equation. Finally, we investigate the limit of th…
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…
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…
We study ageing during surface growth processes described by the one-dimensional Kardar-Parisi-Zhang equation. Starting from a flat initial state, the systems undergo simple ageing in both correlators and linear responses and its dynamical scaling is characterised by the ageing exponents a=-1/3, b=-2/3, lambda_C=lambda_R=1 and z=3/2. The form of the autoresponse scaling function is well described by the recently constructed logarithmic extension of local scale-invariance.
We have studied the Kardar-Parisi-Zhang equation in the strong coupling regime in the mode-coupling approximation. We solved numerically in dimension d=1 for the correlation function at wavevector k. At large times t we found the predicted stretched exponential decay consistent with our previous saddle point analysis in [Phys. Rev. E {\bf 63}, 057103 (2001)], but we also observed that the decay to zero occurred in an unexpected oscillatory way. We have compared the results from mode-coupling for the scaling functions with the recent exact results from Praehofer and Spohn [cond-mat/0101200] for…
The probabilities $P_\pm(t_0,t)$ that a growing Kardar-Parisi-Zhang interface remains above or below the mean height in the time interval $(t_0, t)$ are shown numerically to decay as $P_\pm \sim (t_0/t)^{θ_\pm}$ with $θ_+ = 1.18 \pm 0.08$ and $θ_- = 1.64 \pm 0.08$. Bounds on $θ_\pm$ are derived from the height autocorrelation function under the assumption of Gaussian statistics. The autocorrelation exponent $\bar λ$ for a $d$--dimensional interface with roughness and dynamic exponents $β$ and $z$ is conjectured to be $\bar λ= β+ d/z$. For a recently proposed discretization of the KPZ equation …
This is the article with the same title which is scheduled to appear in the January 2022 issue of the AMS Notices, with additional references which could not be provided in the accepted version due to space constraints. The figures in this article were made collaboratively with Milind Hegde.
The Kardar-Parisi-Zhang model of non-equilibrium critical behaviour (kinetic surface roughening) with turbulent motion of the environment taken into account is studied by the field theoretic renormalization group approach. The turbulent motion is described by the stochastic Navier-Stokes equation with the random stirring force whose correlation function includes two terms that allow one to account both for a turbulent fluid and for a fluid in thermal equilibrium. The renormalization group analysis performed in the leading order of perturbation theory (one-loop approximation) reveals six possib…
We study the scaling regimes for the Kardar-Parisi-Zhang equation with noise correlator R(q) ~ (1 + w q^{-2 ρ}) in Fourier space, as a function of ρand the spatial dimension d. By means of a stochastic Cole-Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d - d_c) expansion. We also argue that there is a intriguing possibility that the rough phases above and below the lower critical dimension d_c = 2 (1 + ρ) are genuinely different which could lead to a re-interpretation of results in the literature.
Using the weak-noise theory, we evaluate the probability distribution $\mathcal{P}(H,t)$ of large deviations of height $H$ of the evolving surface height $h(x,t)$ in the Kardar-Parisi-Zhang (KPZ) equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height $H$ at time $t$. We argue that the tails of $\mathcal{P}$ behave, at arbitrary time $t>0$, and in a proper moving frame, as $-\ln \mathcal{P}\sim |H|^{5/2}$ and $\sim |H|^{3/2}$. The $3/2$ tail coincides with the asymptotic of the Gaussian orthogonal…
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…
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 …
The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar Ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions were evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common …
Kardar-Parisi-Zhang interface depinning with quenched noise is studied in an ensemble that leads to self-organized criticality in the quenched Edwards-Wilkinson (QEW) universality class and related sandpile models. An interface is pinned at the boundaries, and a slowly increasing external drive is added to compensate for the pinning. The ensuing interface behavior describes the integrated toppling activity history of a QKPZ cellular automaton. The avalanche picture consists of several phases depending on the relative importance of the terms in the interface equation. The SOC state is more comp…
