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 here to show that such results are simple consequences of boundary effects, inherent to systems defined on CTs. In fact, I demonstrate that the anomalous boundary of the CT leads the growing surfaces to develop curved shapes, which explains the strange behaviors previously found for these systems, once the global "roughness" were analyzed for non-flat surfaces in the study above. Importantly, by measuring the height fluctuations at the central site of the CT, which can be seen as an approximation for the Bethe lattice, smooth surfaces are found for both EW and KPZ classes, consistently with the behavior expected for growing systems in dimensions $d \geqslant d_u$. Interesting features of the 1-pt height fluctuations, such as the possibility of non-saturation in the steady state regime, are also discussed for substrates in general.
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…
In [arXiv:0804.3035] we studied an interacting particle system which can be also interpreted as a stochastic growth model. This model belongs to the anisotropic KPZ class in 2+1 dimensions. In this paper we present the results that are relevant from the perspective of stochastic growth models, in particular: (a) the surface fluctuations are asymptotically Gaussian on a sqrt(ln(t)) scale and (b) the correlation structure of the surface is asymptotically given by the massless field.
The short time behavior of the 1+1 dimensional KPZ growth equation with a flat initial condition is obtained from the exact expressions of the moments of the partition function of a directed polymer with one endpoint free and the other fixed. From these expressions, the short time expansions of the lowest cumulants of the KPZ height field are exactly derived. The results for these two classes of cumulants are checked in high precision lattice numerical simulations. The short time limit considered here is relevant for the study of the interface growth in the large diffusivity/weak noise limit, …
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 generalize the surface growth model of Gates and Westcott to arbitrary inclination. The exact steady growth velocity is of saddle type with principal curvatures of opposite sign. According to Wolf this implies logarithmic height correlations, which we prove by mapping the steady state of the surface to world lines of free fermions with chiral boundary conditions.
We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painleve II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode c…
Three models from statistical physics can be analyzed by employing space-time determinantal processes: (1) crystal facets, in particular the statistical properties of the facet edge, and equivalently tilings of the plane, (2) one-dimensional growth processes in the Kardar-Parisi-Zhang universality class and directed last passage percolation, (3) random matrices, multi-matrix models, and Dyson's Brownian motion. We explain the method and survey results of physical interest.
We propose a mean field theory for interfaces growing according to the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. The mean field equations are formulated in terms of densities at different heights, taking surface tension and the influence of the nonlinear term in the KPZ equation into account. Although spatial correlations are neglected, the mean field equations still reflect the spatial dimensionality of the system. In the special case of Edwards-Wilkinson growth,our mean field theory correctly reproduces all features. In the presence of a nonlinear term one observes a crossover to…
Stationary states in KPZ type growth have interesting short distance properties. We find that typically they are skewed and lack particle-hole symmetry. E.g., hill-tops are typically flatter than valley bottoms, and all odd moments of the height distribution function are non-zero. Stationary state skewness can be turned on and off in the 1+1 dimensional RSOS model. We construct the exact stationary state for its master equation in a 4 dimensional parameter space. In this state steps are completely uncorrelated. Familiar models such as the Kim-Kosterlitz model lie outside this space, and their …
Control of generically scale-invariant systems, i.e., targeting specific cooperative features in non-linear stochastic interacting systems with many degrees of freedom subject to strong fluctuations and correlations that are characterized by power laws, remains an important open problem. We study the control of surface roughness during a growth process described by the Kardar--Parisi--Zhang (KPZ) equation in $(1+1)$ dimensions. We achieve the saturation of the mean surface roughness to a prescribed value using non-linear feedback control. Numerical integration is performed by means of the pseu…
We introduce a non-perturbative renormalization approach which identifies stable fixed points in any dimension for the Kardar-Parisi-Zhang dynamics of rough surfaces. The usual limitations of real space methods to deal with anisotropic (self-affine) scaling are overcome with an indirect functional renormalization. The roughness exponent $α$ is computed for dimensions $d=1$ to 8 and it results to be in very good agreement with the available simulations. No evidence is found for an upper critical dimension. We discuss how the present approach can be extended to other self-affine problems.
Scaling of surface fluctuations of polycrystalline CdTe/Si(100) films grown by hot wall epitaxy are studied. The growth exponent of surface roughness and the dynamic exponent of the auto-correlation function in the mound growth regime agree with the values of the Kardar-Parisi-Zhang (KPZ) class. The scaled distributions of heights, local roughness, and extremal heights show remarkable collapse with those of the KPZ class, giving the first experimental observation of KPZ distributions in $2+1$ dimensions. Deviations from KPZ values in the long-time estimates of dynamic and roughness exponents a…
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…
We study a noisy Kuramoto-Sivashinsky (KS) equation which describes unstable surface growth and chemical turbulence. It has been conjectured that the universal long-wavelength behavior of the equation, which is characterized by scale-dependent parameters, is described by a Kardar-Parisi-Zhang (KPZ) equation. We consider this conjecture by analyzing a renormalization-group equation for a class of generalized KPZ equations. We then uniquely determine the parameter values of the KPZ equation that most effectively describes the universal long-wavelength behavior of the noisy KS equation.
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.
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 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 $\varepsilon…
Growth in crystals can be { usually } described by field equations such as the Kardar-Parisi-Zhang (KPZ) equation. While the crystalline structure can be characterized by Euclidean geometry with its peculiar symmetries, the growth dynamics creates a fractal structure at the interface of a crystal and its growth medium, which in turn determines the growth. Recent work (The KPZ exponents for the 2+ 1 dimensions, MS Gomes-Filho, ALA Penna, FA Oliveira; \textit{Results in Physics}, 104435 (2021)) associated the fractal dimension of the interface with the growth exponents for KPZ, and provides expl…
We present numerical Monte Carlo results for the stationary state properties of KPZ type growth in two dimensional surfaces, by evaluating the finite size scaling (FSS) behaviour of the 2nd and 4th moments, $W_2$ and $W_4$, and the skewness, $W_3$, in the Kim-Kosterlitz (KK) and BCSOS model. Our results agree with the stationary state proposed by Lässig. The roughness exponents $W_n\sim L^{α_n}$ obey power counting, $α_n= n α$, and the amplitude ratio's of the moments are universal. They have the same values in both models: $W_3/W_2^{1.5}= -0.27(1)$ and $W_4/W_2^{2}= +3.15(2)$. Unlike in one d…
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.
In the spirit of recent works on topological chaos generated by sequential rotation of infinitely thin stirrers placed in a viscous liquid, we consider the statistical properties of braiding exponent which quantitatively characterizes the chaotic behavior of advected particles in two-dimensional flows. We pay a special attention to the random stirring protocol and study the time-dependent behavior of the variance of the braiding exponent. We show that this behavior belongs to the Kardar-Parisi-Zhang universality class typical for models of nonstationary growth. Using the matrix (Magnus) repres…
The crossover from a Berezinskii--Kosterlitz--Thouless (BKT) rough surface to a Kardar--Parisi--Zhang (KPZ) rough surface on a vicinal surface is studied using the Monte Carlo method in the non-equilibrium steady state in order to address discrepancies between theoretical results and experiments. The model used is a restricted solid-on-solid (RSOS) model with a discrete Hamiltonian without surface or volume diffusion (interface limited growth/recession). The temperature, driving force for growth, system size, and surface slope dependences of the surface width are calculated for vicinal surface…
Long-range spatiotemporal correlations may play important roles in nonequilibrium surface growth process. In order to investigate the effects of long-range temporal correlation on dynamic scaling of growing surfaces, we perform extensive numerical simulations of the (1+1)- and (2+1)-dimensional Kardar-Parisi-Zhang (KPZ) growth system in the presence of temporally correlated noise, and compare our results with previous theoretical predictions and numerical simulations. We find that surface morphologies are obviously affected with long-range temporal correlations, and as the temporal correlation…
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…
We examine height-height correlations in the transient growth regime of the 2+1 Kardar-Parisi-Zhang (KPZ) universality class, with a particular focus on the {\it spatial covariance} of the underlying two-point statistics, higher-dimensional analog of the 1+1 KPZ Class Airy$_1$ process. Making comparison to AFM kinetic roughening data in 2d organic thin films, we use our universal 2+1 KPZ spatial covariance to extract key scaling parameters for this experimental system. Additionally, we explore the i) height, ii) local roughness, and iii) extreme value distributions characteristic of these olig…
For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in 1+1 dimensions, fluctuations grow as t^{1/3} during time t and the correlation length at a fixed time scales as t^{2/3}. In this note we discuss the scale of time correlations. For a representant of the KPZ class, the polynuclear growth model, we show that the space-time is non-trivially fibred, having slow directions with decorrelation exponent equal to 1 instead of the usual 2/3. These directions are the characteristic curves of the PDE associated to the surface's slope. As a consequence, previously proven results for spa…
We study discrete KPZ growth models deposited on square lattice substrates, whose (average) lateral size enlarges as $L= L_0 + ωt^γ$. Our numerical simulations reveal that the competition between the substrate expansion and the increase of the correlation length parallel to the substrate, $ξ\simeq c t^{1/z}$, gives rise to a number of interesting results. For instance, when $γ< 1/z$ the interface becomes fully correlated, but its squared roughness, $W_2$, keeps increasing as $W_2 \sim t^{2αγ}$, as previously observed for 1D systems. A careful analysis of this scaling, accounting for an intrins…
We conjecture the universal probability distribution at large time for the one-point height in the 1D Kardar-Parisi-Zhang (KPZ) stochastic growth universality class, with initial conditions interpolating from any one of the three main classes (droplet, flat, stationary) on the left, to another on the right, allowing for drifts and also for a step near the origin. The result is obtained from a replica Bethe ansatz calculation starting from the KPZ continuum equation, together with a "decoupling assumption" in the large time limit. Some cases are checked to be equivalent to previously known resu…
We explore linear control of the one-dimensional non-linear Kardar--Parisi--Zhang (KPZ) equation with the goal to understand the effects the control process has on the dynamics and on the stationary state of the resulting stochastic growth kinetics. In linear control, the intrinsic non-linearity of the system is maintained at all times. In our protocol, the control is applied to only a small number $n_c$ of Fourier modes. The stationary-state roughness is obtained analytically in the small-$n_c$ regime with weak non-linear coupling wherein the controlled growth process is found to result in Ed…
Flat surface phases are unstable during growth and known to become rough. This does not exclude the possibility that surface reconstruction order persists in rough growing surfaces, in analogy with so-called equilibrium reconstructed rough phases. We investigate this in the context of KPZ type dynamics, using the restricted solid on solid model with negative mono-atomic step energies. Long range reconstruction order is strictly speaking absent in the thermodynamic limit, but the reconstruction domain walls become trapped at surface ridge lines, and the reconstruction order parameter fluctuates…
We introduce a model of effective conformal quantum field theory in dimension $d=1+1$ coupled to stochastic noise, where Kardar-Parisi-Zhang (KPZ) class fluctuations can be observed. The analysis of the quantum dynamics of the scaling operators reduces to the study of random trajectories in a random environment, modeled by Brownian vector fields. We use recent results on random walks in random environments to calculate the time-dependent entanglement entropy of a subsystem interval, starting from a factorized state. We find that the fluctuations of the entropy in the large deviation regime are…
Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrate…
We obtain several exact results for universal distributions involving the maximum of the Airy$_2$ process minus a parabola and plus a Brownian motion, with applications to the 1D Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g. $C_{\infty} = \lim_{t_2/t_1 \to +\infty} \overline{h(t_1) h(t_2)}^c/\overline{h(t_1)^2}^c$, with $C_{\infty} \approx 0.623$, as well as conditional moments, which quantify ergodicity breaking in the time evoluti…
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.
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…
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…
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, $ΔW^2 \sim s^c t^{-γ}$, 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 co…
I report on an extensive numerical investigation of various discrete growth models describing equilibrium and nonequilibrium interfaces on a substrate of a finite Bethe lattice. An unusual logarithmic scaling behavior is observed for the nonequilibrium models describing the scaling structure of the infinite dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This gives rise to the classification of different growing processes on the Bethe lattice in terms of logarithmic scaling exponents which depend on both the model and the coordination number of the underlying lattice. T…
We investigate an off-lattice Eden model where the growth of new cells is performed with a probability dependent on the availability of resources coming externally towards the growing aggregate. Concentration of nutrients necessary for replication is assumed to be proportional to the voids connecting the replicating cells to the outer region, introducing therefore an nonlocal dependence on the replication rule. Our simulations point out that the Kadar-Parisi-Zhang (KPZ) universality class is a transient that can last for long periods in plentiful environments. For conditions of nutrient scarci…
We consider the fluctuations in the stochastic growth of a one-dimensional interface of height $h(x,t)$ described by the Kardar-Parisi-Zhang (KPZ) universality class. We study the joint probability distribution function (JPDF) of the interface heights at two times $t_1$ and $t_2>t_1$, with droplet initial conditions at $t=0$. In the limit of large times this JPDF is expected to become a universal function of the time ratio $t_2/t_1$, and of the (properly scaled) heights $h(x,t_1)$ and $h(x,t_2)$. Using the replica Bethe ansatz method for the KPZ equation, in [J. Stat. Mech. (2017) 053212] we o…
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, we investigate here several 1D 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 $L_0 \gg 1$ 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, $L_0$ …
We consider some models in the Kardar-Parisi-Zhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion process. For these models, in the limit of large time t, universality of fluctuations has been previously obtained. In this paper we consider the convergence to the limiting distributions and determine the (non-universal) first order corrections, which turn out to be a non-random shift of order t^{-1/3} (of order 1 in microscopic units). Subtracting this deterministic correction, the convergence is then of order t^{-2/3}. We also deter…
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.
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 study the $(1+1)$-dimensional Kardar-Parisi-Zhang (KPZ) interfaces growing inward from ring-shaped initial conditions, experimentally and numerically, using growth of a turbulent state in liquid-crystal electroconvection and an off-lattice Eden model, respectively. To realize the ring initial condition experimentally, we introduce a holography-based technique that allows us to design the initial condition arbitrarily. Then, we find that fluctuation properties of ingrowing circular interfaces are distinct from those for the curved or circular KPZ subclass and, instead, are characterized by t…
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…
Obtaining the exact multi-time correlations for one-dimensional growth models described by the Kardar-Parisi-Zhang (KPZ) universality class is presently an outstanding open problem. Here, we study the joint probability distribution function (JPDF) of the height of the KPZ equation with droplet initial conditions, at two different times $t_1<t_2$, in the limit where both times are large and their ratio $t_2/t_1$ is fixed. This maps to the JPDF of the free energies of two directed polymers with two different lengths and in the same random potential. Using the replica Bethe ansatz (RBA) method, w…
We study a restricted solid-on-solid (RSOS) model involving deposition and evaporation with probabilities p and 1-p, respectively, in one-dimensional substrates. It presents a crossover from Edwards-Wilkinson (EW) to Kardar-Parisi-Zhang (KPZ) scaling for p~0.5. The associated KPZ equation is analytically derived, exhibiting a coefficient lambda of the nonlinear term proportional to q=p-1/2, which is confirmed numerically by calculation of tilt-dependent growth velocities for several values of p. This linear λ-q relation contrasts to the apparently universal parabolic law obtained in competitiv…
We simulated a growth model in 1+1 dimensions in which particles are aggregated according to the rules of ballistic deposition with probability p or according to the rules of random deposition with surface relaxation (Family model) with probability 1-p. For any p>0, this system is in the Kardar-Parisi-Zhang (KPZ) universality class, but it presents a slow crossover from the Edwards-Wilkinson class (EW) for small p. From the scaling of the growth velocity, the parameter p is connected to the coefficient of the nonlinear term of the KPZ equation, lambda, giving lambda ~ p^gamma, with gamma = 2.1…
We study the atypically large deviations of the height $H \sim {\cal O}(t)$ at the origin at late times in $1+1$-dimensional growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. We present exact results for the rate functions for the discrete single step growth model, as well as for the continuum KPZ equation in a droplet geometry. Based on our exact calculation of the rate functions we argue that models in the KPZ class undergo a third order phase transition from a strong coupling to a weak coupling phase, at late times.
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.
The static and dynamic roughenings of a growing crystalline facet is studied where the growth mechanism is controlled by a restricted-curvature (RC) geometry. A continuum equation, in analogy with the Kardar-Parisi-Zhang (KPZ) equation is considered for the purpose. It is shown here that although the growth process begins with a RC geometry, a structural phase transition occurs from the restricted-curvature phase to the KPZ phase. An estimation of the corresponding critical temperature is given here. Calculations on the static phase transition give results along the same line as existing predi…
The statistics of the average height fluctuation of the one-dimensional Kardar-Parisi-Zhang(KPZ)-type surface is investigated. Guided by the idea of local stationarity, we derive the scaling form of the characteristic function in the early-time regime, $t\ll N^{3/2}$ with $t$ time and $N$ the system size, from the known characteristic function in the stationary state ($t\gg N^{3/2}$) of the single-step model derivable from a Bethe Ansatz solution, and thereby find the scaling properties of the cumulants and the large deviation function in the early-time regime. These results, combined with the…
We study the dynamics of a growing crystalline facet where the growth mechanism is controlled by the geometry of the local curvature. A continuum model, in (2+1) dimensions, is developed in analogy with the Kardar-Parisi-Zhang (KPZ) model is considered for the purpose. Following standard coarse graining procedures, it is shown that in the large time, long distance limit, the continuum model predicts a curvature independent KPZ phase, thereby suppressing all explicit effects of curvature and local pinning in the system, in the "perturbative" limit. A direct numerical integration of this growth …
