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Dynamical universality classes of simple growth and lattice gas models Jeffrey Kelling (2,3), Géza Ódor (1) and Sibylle Gemming (3,4) (1) Institute of Technical Physics and Materials Science, Centre for Energy Research of the Hungarian Academy of Sciences P.O.Box 49, H-1525 Budapest, Hungary (2) Department of Information Services and Computing, Helmholtz-Zentrum Dresden-Rossendorf P.O.Box 51 01 19, 01314 Dresden, Germany (3) Institute of Ion Beam Physics and Materials Research Helmholtz-Zentrum Dresden-Rossendorf 7 P.O.Box 51 01 19, 01314 Dresden, Germany 1 (4) Institute of Physics, TU Chemnitz 0 09107 Chemnitz, Germany 2 n Large scale, dynamical simulations have been performed for the two dimensional octahedron a model,describingKardar–Parisi–Zhang(KPZ)fornonlinear,orEdwards–Wilkinson(EW)forlinear J surfacegrowth. Theautocorrelationfunctionsoftheheightsandthedimerlatticegasvariablesare 3 determined with high precision. Parallel random-sequential (RS) and two-sub-lattice stochastic 1 dynamics (SCA) have been compared. The latter causes a constant correlation in the long time limit,butaftersubtractingitonecanfindthesameheightfunctionsasincaseofRS.Ontheother ] handtheorderedupdatealtersthedynamicsofthelatticegasvariables,byincreasing(decreasing) h the memory effects for nonlinear (linear) models with respect to RS. Additionally, we support the c KPZ ansatz and the Kallabis-Krug conjecture 𝜆 = 𝐷 in 2+1 dimensions and provide a precise e 𝐶 growthexponentvalue𝛽 =0.2414(2). Weshowtheemergenceoffinitesizecorrections,whichoccur m much earlier than the height saturation. - t a PACSnumbers: 05.70.Ln,05.70.Np,82.20.Wt t s . at I. INTRODUCTION amplitude, relatedtothetemperatureintheequilibrium m system, averaged over the distribution. The 𝜆 = 0, lin- - Nonequilibrium systems are known to exhibit dynam- earequationdescribestheEdwards–Wilkinson(EW)[17] d ical scaling, when the correlation length diverges as 𝜉 surface growth, an exactly solvable equilibrium system. n 𝑡1/𝑧, characterized by the exponent 𝑧. Simplest mode∝ls Severaldiscretizedversionshavebeenstudied[2,7,18]. o are driven lattice gases (DLG) [1], which in certain cases Inthesemodelsthemorphologyofasurfaceoflinearsize c [ can be mapped onto surface growth [2, 3]. Therefore, 𝐿 is usually described by the squared interface width understanding DLGs, which is far from being trivial due 1 𝐿 𝐿 to the broken time reversal symmetry [4], and is pos- 1 1 2 v 𝑊2(𝐿,𝑡)= ℎ2 (𝑡) ℎ (𝑡) . (2) 8 sible mostly by numerical simulations only, sheds some 𝐿2 𝑖,𝑗 − 𝐿 𝑖,𝑗 3 light on the corresponding interface phenomena [5]. The ∑︁𝑖,𝑗 (︁ ∑︁𝑖,𝑗 )︁ 6 simplestexampleistheasymmetricsimpleexclusionpro- Intheabsenceofanycharacteristiclengthsimplegrowth 3 cess (ASEP) of particles [6], in which particles and holes processes are expected to be scale-invariant 0 can be mapped onto binary surface slopes [7, 8] and the . 1 corresponding continuous model can be described by the 𝑊(𝐿,𝑡) 𝐿𝛼𝑓(𝑡/𝐿𝑧) , (3) 0 Kardar–Parisi–Zhang (KPZ) equation [9] ∝ 7 1 𝜕𝑡ℎ(x,𝑡)=𝜎 2ℎ(x,𝑡)+𝜆( ℎ(x,𝑡))2+𝜂(x,𝑡) , (1) with the universal scaling function 𝑓(𝑢): ∇ ∇ : v where the scalar field ℎ(x,𝑡) is the height, progressing 𝑢𝛽 if 𝑢 1 Xi in the 𝐷 dimensional space relative to its mean position, 𝑓(𝑢)∝ const. if 𝑢≪1 (4) that moves linearly with time 𝑡. This equation was in- {︂ ≫ ar spired in part by the stochastic Burgers equation [10] Here 𝛼 is the roughness exponent in the stationary and can describe the dynamics of simple growth pro- regime, when 𝜉 has exceeded 𝐿 and 𝛽 is the growth ex- cesses in the thermodynamic limit [11], randomly stirred ponent, describing the intermediate time behavior. The fluids [12], directed polymers in random media [13], dis- dynamical exponent 𝑧 can be expressed as the ratio of sipativetransport[14,15], andthemagneticfluxlinesin the growth exponents: superconductors [16]. In case of surface growth 𝜎 rep- resents a surface tension, competing with the nonlinear 𝑧 =𝛼/𝛽 (5) up–down anisotropy of strength 𝜆 and a zero mean val- ued Gaussian white noise. This field exhibits the vari- In two as well as in higher dimensions not much is ance 𝜂(x,𝑡)𝜂(x,𝑡) = 2Γ𝛿𝐷(x x)(𝑡 𝑡), with an knownanalyticallyaboutthescalingbehavioroftheKPZ ′ ′ ′ ′ ⟨ ⟩ − − 2 equation (1). The Galilean symmetry [12] implies the surface height ℎ(𝑡,r). In the latter case, for 𝑡 = 𝑠 one scaling relation: finds: 𝐶 (𝑡,𝑠)= ℎ(𝑡,r)ℎ(𝑡,r) ℎ(𝑡,r) ℎ(𝑡,r) ℎ ⟨ ⟩−⟨ ⟩⟨ ⟩ 2=𝛼+𝑧 =𝛼(1+1/𝛽) (6) = ℎ2(𝑡,r) ℎ(𝑡,r) ℎ(𝑡,r) −⟨ ⟩⟨ ⟩ Thisequationrelatestheroughnesstothedynamicaland =⟨︀𝑊2(𝐿 ⟩︀ ,𝑡) 𝑠−𝑏ℎ 𝑓𝐶(1) . →∞ ∝ · growth exponents, allowing one to check numerically ob- This implies the relation tained estimates for consistency. Apart from the exponents, the shapes of the rescaled 𝑏ℎ = 2𝛽 , (14) − width and height distributions of the interfaces Ψ (𝜙 ) 𝐿 𝐿 which must be satisfied in the 𝐿 and 𝑠 limit. were shown to be universal in KPZ models [19, 20]. In →∞ →∞ We have also calculated the auto-correlation of the slope fact many people define the universality classes by these (density) variables quantities, which can be obtained exactly in one dimension[21,22]. Here,𝜙𝐿denotestheinterfaceobservablein 𝐶𝑠(𝑡,𝑠) = (𝑛(𝑡;⃗𝑟) 𝑛(𝑡;⃗𝑟) )(𝑛(𝑠;⃗𝑟) 𝑛;⃗𝑟(𝑠) ) question,𝑊2 orℎ,inasystemoflinearsize𝐿. Thenon- ⟨ −⟨ ⟩ −⟨ ⟩ ⟩ = 𝑛(𝑡;⃗𝑟)𝑛(𝑠;⃗𝑟) 𝑛(𝑡;⃗𝑟) 𝑛(𝑠;⃗𝑟) rescaledprobabilitydistributionsaredenotedby𝑃 (𝜙 ) ⟨ ⟩−⟨ ⟩⟨ ⟩ 𝐿 𝐿 𝑡 aangedstahse:ir moments are defined via the distribution aver- = 𝑠−𝑏𝑠𝑓𝐶′ 𝑠 , (15) (︂ )︂ where 𝑛(𝑡;⃗𝑟) is the occupancy of the sites. How- Φ𝑛[𝜙 ]= ∞(𝜙 𝜙 )𝑛𝑃 (𝜙 )d𝜙 , (7) ever, 𝐶𝑠(𝑡,𝑠) decays much faster than the height auto- 𝐿 𝐿 𝐿−⟨ 𝐿⟩ 𝐿 𝐿 𝐿 correlatorandobtainingreasonablesignal/noiseratiore- ∫︁ 0 quires much higher statistics. A dynamic, perturbative renormaliztion group (RG) Two standard measures of the shape, the skewness analysisofKPZ[24,25]suggestedthattheshortandthe longtimescalingbehaviorofthecorrelationfunctionare 𝑆𝐿[𝜙𝐿]=⟨Φ3𝐿[𝜙𝐿]⟩/⟨Φ2𝐿[𝜙𝐿]⟩3/2 (8) identicalanddeducedascalingrelationfortheshort-time exponent and the kurtosis 𝜃 =(𝐷+4)/𝑧 2 . (16) − 𝑄𝐿[𝜙𝐿]=⟨Φ4𝐿[𝜙𝐿]⟩/⟨Φ2𝐿[𝜙𝐿]⟩2−3 , (9) Since 𝜃 = 𝜆𝐶/𝑧+2𝛽, relation (16) holds in the exactly solvable 1+1 dimensional case. In 𝐷 2 dimensions are calculated usually, often in the steady state. The perturbative RG can’t access the strong≥coupling KPZ universal, rescaled forms are: fixed point [26], thus the validity of this law should be tested by precise exponent estimates. Ψ [𝑊2(𝐿)]= 𝑊2(𝐿) 𝑃 (𝑊2(𝐿)/ 𝑊2(𝐿) ) (10) A conjecture based on a purely geometric argument, 𝐿 𝐿 ⟨ ⟩ ⟨ ⟩ advanced by [27], which can also be deduced from the for the width and scaling relation (16) claims 𝐷 = 𝜆𝐶 in any dimensions. While it is certainly true in one dimension, we will also Ψ [ℎ (𝑟)]=𝐿𝛼𝑃 (ℎ (𝑟)/𝐿𝛼) (11) provide numerical evidence for the validity in 𝐷 =2. 𝐿 𝐿 𝐿 𝐿 In aging systems a similar scaling form is expected for for the surface height. Note, that ℎ Φ0[ℎ ] 0 in the autoresponse function of the field 𝜑: ⟨ 𝐿⟩ ≡ 𝐿 𝐿 ≡ the co-moving frame of the surface. 𝛿 𝜑(𝑡) 𝑡 While many systems are described by a single dynam- 𝑅(𝑡,𝑠)= ⟨ ⟩ =𝑠−1−𝑎𝑓𝑅 (17) 𝛿𝑗(𝑠) 𝑠 ical length scale, aging ones are best characterized by ⃒𝑗=0 (︂ )︂ ⃒ two-time quantities, such as the dynamical correlation where 𝑗 is the external c⃒onjugate to 𝜑 and 𝑎 denotes ⃒ andresponsefunctions[23]. Intheagingregime: 𝑠 𝜏m the so-called aging exponent 𝑎. The universal scaling ≫ a𝑠nids t𝑡h−e𝑠s≫tar𝜏tmt,imweh,ewreh𝜏emn itshae msniacproshscootpiiscttaimkeens,coanleeaenxd- (fu𝑡/n𝑠c)t−io𝜆n𝑅/e𝑧xwhiibthitsthtehaeutaosryemsppotnosteicexbpeohnaevniotr𝜆𝑅𝑓𝑅.(I𝑡n/𝑠e)qu∼i- pects the following law for the autocorrelation function librium 𝜆 = 𝜆 and 𝑎 = 𝑏 due to the fluctuation- 𝐶 𝑅 dissipation (FD) symmetry [28]. In nonequilibrium sys- 𝐶(𝑡,𝑠)= 𝜑(𝑡,r)𝜑(𝑠,r) 𝜑(𝑡,r) 𝜑(𝑠,r) (12) tems these exponents can be completely indepenedent. ⟨ ⟩−⟨ ⟩⟨ ⟩ 𝑠−𝑏(𝑡/𝑠)−𝜆𝐶/𝑧 , (13) Therefore, we shall determine them one-by-one and in- ∝ vestigateifsomeextendedFDrelationmayoccuramong here denotes averaging over the sites and independent them. This will be done using our very recent aging re- ⟨⟩ samples, 𝜆 is the autocorrelation and 𝑏 is the aging sponse exponents [29], which were determined to test 𝐶 exponent. Thefunction𝜑denotesthemeasuredquantity, the validity of the logarithmic extension of local scale- whichcanbetheparticledensityofthelatticegasorthe invariance (LSI) [28] proposed by [30]. 3 II. MODELS AND SIMULATION ALGORITHMS Discrete models set up for KPZ have been studied a lot in the past decades [2, 7, 18]. A mapping between KPZsurfacegrowthintwodimensionsanddrivenlattice gases has been advanced in [31, 32] an extension of the "rooftop" model of [7, 8]. We call it octahedron model, which is characterized by binary slope variables 𝜎 at 𝑥/𝑦 the edges connecting the vertexes of octahedra [33]. The 𝜎 take the values 0 or 1 to encode down or up slopes, 𝑥/𝑦 respectively. Thus deposition or removal of octahedra corresponds to a stochastic cellular automaton with the simple Kawasaki update rules 0 1 𝑝 1 0 (cid:10) , (18) 0 1 𝑞 1 0 (︂ )︂ (︂ )︂ where 𝑝 and 𝑞 denote the acceptance probabilities. Pro- jecting the edges onto a plane yields a square lattice of slopes, which can then be considered as occupancy variables. This maps the octahedron model onto self- reconstructing dimers following an oriented migration along the bisection of the 𝑥 and 𝑦 directions of the sur- FIG.1: Schematicsofthetwosub-latticeSCAupdatesofthe face (see Fig. in [32]). In this picture the surface heights dimer lattice gas model. Circles: empty sites (down slopes), mustbedefinedrelativetoareferencepointℎ =0and bullets: filled sites (up slopes). Solid, black lines denote ar- 1,1 can be reconstructed from the slope variables as eas, wheretherule(18)isapplied at𝑡odd, whiledashedred lines encircle areas for update at even 𝑡 timesteps. Diago- 𝑖 𝑗 nal, dashed lines are parallel with the 𝑥 and 𝑦 axis and are ℎ𝑖,𝑗 = [2𝜎𝑥(𝑙,1) 1]+ [2𝜎𝑦(𝑖,𝑘) 1] . (19) projection of the octahedron edges. − − 𝑙=1 𝑘=1 ∑︁ ∑︁ Discrete surface and DLG models usually apply ran- ing laws were performed [5]. For exmple in case of the dom sequential dynamics. On the other hand in certain interface width growth we used cases synchronous, so called stochastic cellular automa- 𝑡 𝑡 ln𝑊(𝐿 ,𝑡 ) ln𝑊(𝐿 ,𝑡 ) ton(SCA)-likesiteupdatingcanprovetobeuseful,espe- 𝑖 𝑖/2 𝑖 𝑖/2 𝛽 − = →∞ − →∞ . eff cially for simulations on parallel computers. This study (︂ 2 )︂ ln(𝑡𝑖)−ln(𝑡𝑖/2) is based on massively parallel simulations on graphics (20) cards (GPUs) . Synchronous updating in case of one- In our studies the simulation time, measured in Monte dimensional ASEP models has already been investigated Carlo steps (MCS), between two measurements was in- [34–36]. One-point quantities in the bulk, like particle creased exponentially current or surface growth have been shown to exhibit 𝑡 =(𝑡 +10)e𝑚 , (21) the same behavior as in case of random-sequential (RS). 𝑖+1 𝑖 However, n-point correlation functions may be different. using 𝑚 = 0.01 and 𝑡 = 0. Statistical uncertainties 0 Here we extend the parallel two-sublattice scheme de- are provided as 1𝜎–standard errors, defined as Δ1𝜎𝑥 = veloped for ASEP [35] to the two dimensional dimer 𝑥2 𝑥 2/(𝑁 1). ⟨ ⟩−⟨ ⟩ − model as shown on Fig. 1, and compare the dynami- Througout this study we used the implementation of √︀ cal scaling results with those of the RS dynamics. Since the Levenberg–Marquadt algorithm [41, 42] in the gnu- blocks of sites to be updated can be visited in a sequen- plot software [43] for non-linear least squres fitting. tial order within a sub-lattice step, the SCA dynamics For𝑝=𝑞 >0theoctahedronmodeldescribesthesur- can be implemented very efficiently [37], matching per- facegrowthoftheEWequation[32]in2+1dimensions. fectly parallel processors of GPU architectures [38]. In this case the autocorrelation function of heights has Performing RS simulations on GPUs is less straight- been derived [44, 45]: forward, because unwanted correlations may be intro- 𝑡+𝑠 duced [39]. In order to eliminate these and to achieve 𝐶EW =𝑐 ln , (22) h 0 𝑡 𝑠 results as close to really sequential simulations as possi- (︂ − )︂ ble, we use a new domain decomposition (DD) scheme, where 𝑐 is a model-dependent constant. This function 0 whichwecallDTrDB.Detailsofthenewimplementation approaches 0 for 𝑡 𝑠 as a power law (PL) with the ex- ≫ are documented elsewhere [40]. ponent𝜆 /𝑧 =1,where𝑧 =2. InSect.IIIB2 𝐶,h,EW EW EW In order to estimate the asymptotic values of different weshallreproducethisresultnumericallyasatestofour exponents for 𝑡 , local slope analyses of the scal- simulations. → ∞ 4 III. RESULTS 𝑝=0.0625 𝑝=0.125 101 𝑝=0.25 𝑝=0.40 Extensivedynamicalsimulationswereperformedusing 𝑝=0.50 bothRSandSCAupdatingschemes. Toavoidfinite-size 𝑝=0.60 effects we considered large systems with lateral sizes of 𝐿=216. UsingparallelizedRSsimulationswithDD,de- 𝑝=0.80 noted as DTrDB, raises the question whether the results 𝑊 𝑝=0.95 depend on the chosen domain sizes. This is addressed herebypresentingresultsfortwodifferentDDconfigura- tions, denoted as TC=1,1 and TC=2,2. Domains contain 100 8 8 sites in the first case and 16 16 in the second. In (a) × × bothcasestheprobabilityforparticledepositionis𝑝=1. InSCAsimulationsthedepositionprobabilitymustbe 𝑝<1 in order to allow stochastic noise. We investigated 101 103 105 three cases: 𝑝=0.5,0.75 and 0.95 in depth. While KPZ 𝑡 𝑝e𝑝 runswereperformedwithoutremovals: 𝑞 =0,intheEW · growth we applied 𝑝=𝑞 =1 for RS and 𝑝=𝑞 =0.5 for 0.242 SCA. The roughness scaling of the interface width is ana- lyzedinsectionIIIA.Thisisfollowedbyautocorrelation andagingstudiesoftheheightaswellaslattice-gasvari- 0.240 ables in section IIIB. ff DTrDB, TC=2,2 e A. Roughness scaling 𝛽 0.238 SCA, 𝑝 = 0.50, 𝐿 = 217 SCA, 𝑝 = 0.95, 𝐿 = 216 To compare numerical results coming from different SCA, 𝑝 = 0.75, 𝐿 = 216 upates we determined experimentally a scaling function 0.236 SCA, 𝑝 = 0.50, 𝐿 = 216 (b) 𝑓(𝑡,𝑝), that provides collapses of 𝑊(𝐿,𝑓(𝑡,𝑝)) for different dynmics. In case or RS dynamics this function is linear 𝑓RS(𝑡,𝑝) 𝑝. Since the 𝑝 0 limit of the SCA 10−6 10−5 10−4 10−3 10−2 ∝ → corresponds to RS updating, we tried to extend the lin- 1/𝑡 ear form analytically. A smaller survey study of SCA for a larger number of different 𝑝 0.95 values was used ≤ to obtain this function numerically and resulted in a the FIG. 2: Width-scaling under SCA dynamics. (a) Width following nonlinear extension: curves collapsed over 𝑝 by rescaling tim̃︀e as ̃︀𝑡 = 𝑡·𝑝e𝑝. For comparison the DTrDB TC=2,2 result is also shown (black 𝑓SCA(𝑡,𝑝)=𝑡(𝑝)=𝑡 𝑝e𝑝 . (23) dashed line). (b) Effective scaling exponents under SCA dy- · namics for 𝑝 = 0.95 (𝑛 ≥ 2254), 𝑝 = 0.75 (𝑛 ≥ 6430) and The speedup with respect ̃︀to linear function of RS can 𝑝 = 0.5 (𝑛 ≥ 373, 𝑛 ≥ 3062). RS data is shown for com- be understood as follows. A dimer, that was moved at parison (𝑛≥708). Propagated 1𝜎 error bars are attached to a given timestep becomes the target of another update theeffectiveexponents,mergingintoanerror-corridoratlate at the next sub-lattice step in the 𝑝 1, 𝑞 = 0 case. times due to the dense placing of points. → This is more effective than random sequential updating. Therefore, the roughness of the KPZ grows faster than in case of RS dynamics. One can test this function by plateau values, yields 𝛽 = 0.24146(1) and DTrDB,TC=1,1 observing a reasonably collapse on Fig. 2(a) for different 𝛽 = 0.24139(1). The case TC=1,1 was ex- DTrDB,TC=2,2 𝑝-s of SCA as compared to the RS results. cluded from figure 2(b) to reduce cluttering. It is Figure 2(b) shows the effective scaling exponents of presented separately in figure 4(a) and shows slightly 𝛽 (20) for SCA and RS simulations as the function of stronger corrections, which we ignored by assuming a the rescaled time variable. Most notably, the 𝛽 ex- constant plateau. This causes slightly different average eff ponent results exhibit slightly shifted plateaus for al- values and uncertainty beyond the pure statistical error. most two decades in time, but the difference lies well Therefore,weprovidetheunifiedestimate𝛽 =0.2414(2), within the error-margin of our best bublished result with an error margin of about the same size as the 1𝜎- 𝛽 =0.2415(15) [46]. error bars attached to the effective exponents at late eff a. Random-sequential The pronounced plateau vis- times. ible for DTrDB, TC=2,2, suggests that corrections at b. Stochastic cellular automaton Like in the RS these late times are small. Simple averaging of the case, there is a plateau spanning almost two decades 5 in the 𝐿 = 216 datasets, suggesting significantly different scaling exponents, which depend on 𝑝: 𝛽 = (a) 𝑝=0.95 0.4 0.24079(1) and 𝛽 = 0.24122(1). While the latter o 𝑝=0.5 i falls within the error margin of the value derived from t a RS runs, the former does not. The spread of these val- R ues Δ𝛽SCA = 0.00043(2) is marginally significant, even t0.3 n though corrections have not been considered. The effec- a 𝑆(𝑡), 𝐿=212 tiveexponentsoftheSCArunswith𝑝=0.75showinter- l − u 𝑆(𝑡), 𝐿=216 mediate behavior: while they are close to 𝛽eff,𝑝=0.5(1/𝑡), m0.2 − ̃︀ tthheeycuternved𝛽downward(1a/f𝑡t)e.r 1/𝑡 (cid:38) 1×10−4 and approã︀ch Cu −𝑄𝑆(𝑡∞)̃︀, 𝐿==0.422166596 eff,𝑝=0.95 The possibility, that SCÃ︀simulations exhibit differ- 0.1 𝑄∞̃︀ =0.350619 ent growth exponents d̃︀epending on 𝑝 < 1 raises doubts about correct modelling KPZ surface growth. In this 101 102 103 104 105 106 case, the asymptotic scaling exponents obtained by SCA woulddifferfromtheactualKPZvalueandconvergeonly 𝑡 [MCS] in the RS-limit 𝑝 1/𝑉. The plots also show a break → down of 𝛽 at late times. Such behavior can be at- o eff i tributed to the onset of the steady state, which seems t ̃︀ a to be unlikely: 𝜉 𝑡𝑧 𝐿 would be reached about one R10 1 ∝ ∼ − decade later than the left end of the displayed plot. An- t other explanation might be the possibility of blockades an10 2 iwnhitchhesdtiamrtertolamttoicvee-gtahsroduugehttohethseysctoermrelaastewdauvepsd,aatensd, mul − ∼𝑡−0.6 10 3 slow down the growth of surface roughness. This effect u − candependon𝑝,butshouldbeindependentof𝐿,except C ̃︀ for very small systems, where real finite size effects set o10 4 𝑆(𝑡) 𝑆 , 𝐿=212 t − − ∞ in before the wave phenomenon could be observed. The ⃒𝑆(𝑡) 𝑆 ⃒, 𝐿=216 presented data suggest scaling of this cut-off with 𝑝, be- rr.10−5 ⃒⃒𝑄(̃︀𝑡)− 𝑄∞⃒⃒, 𝐿=216 (b) cause the kink appears at about the same rescaled time o ⃒ ̃︀ − ∞⃒ 𝑡≈1.7×105MCSinallSCAsimulationsofsize𝐿=216. C 101 ⃒⃒ 1̃︀02 10⃒⃒3 104 105 106 Most importantly, the 𝛽 curve does not exhibit cut- eff ̃︀offintheplateauincaseofourlargestsized𝐿=217data, 𝑡 [MCS] but matches perfectly the RS result. It only shows noise related oscillations within the 1𝜎-error margin. This in- dicates that the cut-off is related to finite sizes that will FIG.3: Skewness𝑆 andkurtosis𝑄ofthedistributionofin- ̃︀ be investigated further in the following section. terfaceheightsinthegrowthregime. Thedatabelongstothe set of SCA runs with 𝑝=0.75, 𝐿=216 (𝑛 ≥6430, SCA,p=0.75 compare figure 2). The skewness for a smaller dataset for 1. Distribution of interface heights in the growth regime 𝐿 = 212 (𝑛SCA,p=0.75,L=212 ≥ 45) is included to illustrate finite-sizebehavior. (a)Cumulantratiosasfunctionsoftime. Thehorizontallinesshowtheobtainedfitparametersforthe Inordertogetinformationabouttheshapesofthedis- asymptotic values, to guide the eye. See text for proper val- tributionoftheSCAinterfaceheightswecalculatedtheir ues with error estimates. (b) Finite-time and finite-size cor- lowest moments. Fig. 3 shows the evolution of the cu- rections to the asymptotic values of the cumulant ratios. mulantratios𝑆 and𝑄, definedbyequations(8)and(9). Thecurvesapproachtheirgrowthregimeasymptoticval- ues, but move away again at late times. These values: 𝑆 depends on the choice 𝑝 ≷ 𝑞 in the simulations, cor- 𝑆 and 𝑄 can be determined by performing a fit of th∞e form ∞ responding to sign(𝜆) of the KPZ equation (1). Panel (b) of Fig. 3 shows the deviations from these asymptotic 𝑅(𝑡)=𝑅 +𝑏𝑅 𝑡𝑐𝑅 , (24) values. The error estimates given above originate from ∞ · this representation: The error is assumed to be by the order of the closest approach of the numerical data to where 𝑅 is a placeholder for 𝑆 õ︀r 𝑄, in the interval: 200 𝑡 200000MCS, which excludes early time os- the asymptotic value. Exponents of (24) are 𝑐𝑆 0.54 ≈− cillat≤ions≤as well as the cut-off at late times, coming and 𝑐𝑄 0.60. The value 𝑐𝑅 0.6 also holds for ≈ − ≈ − a number of other dimensionless cumulant rations, not from 𝜉̃︀ 𝐿. This yields 𝑆h,growth = 0.427(2) and → − displayed here. 𝑄 = 0.351(3), in agreement with literature val- h,growth ues [47, 48] for the KPZ universality class. The sign of After the closest approach to the asymptotic values 6 in the growth regime, 𝑆(𝑡) and 𝑄(𝑡), both, move in runs on the other hand show strong oscillations at early the direction of their respective values in the steady times, caused by the synchronous updates and are not state: 𝑆h,steady 0.26 and̃︀𝑄h,steady ̃︀ 0.13 [37, 49, 50]. expected to be described by the KPZ ansatz. Still, late ≈ ≈ The shape of the distribution of surface heights chang- times, in the interval 1 10 5 1/𝑡 3 10 3, can − − × ≤ ≤ × ing in this way is an indication of finite-size effects be- be fitted well by (26). The case 𝑝 = 0.75 poses some coming relevant at 𝑡fs 3 105MCS. This coincides exceptionhere,duetoitsnon-monotoñ︀ouscharacteristics ≈ × with the time at which the cut-off [66] in 𝛽 (𝑡) was ob- in the apparent cross-over from 𝑝=0.5-like to 𝑝=0.95- eff served in SCA runs̃︀for 𝐿 = 216 (see Fig. 2(b)). Hence like behavior. it becomes clear, that this change in 𝛽eff is ̃︀caused by A more quantitative summary is provided in Tab. I, finite-size effects. In Fig. 3(b) we also plotted 𝑆(𝑡) of a which lists the reduced sums of residuals 𝜒 to quan- red smaller system, for which the steady state is reached at tify agreement between the fitting model and the data. 𝑡<4 105MCS 212𝑧. ̃︀ Eq. (26) describes all datasets the best, if two terms × ≃ 𝑡 2𝛽 and 𝑡 4𝛽 (𝑁 = 2) are included, except SCA − − ∝ ∝ ̃︀ with 𝑝 = 0.75, which requires more free parameters to 2. KPZ ansatz for the growth regime app̃︀roximate it’̃︀s more complicated form at late times. Where the KPZ ansatz does indeed apply, fits of the AnalyticalandnumericalinvestigationsofKPZmodels more general models (25) should not show increased in 1+1 dimensions found that finite-time corrections to agreementwiththedata. Thetableshowsthemtobeless ℎ(𝑡)tooktheform 𝑡 𝛽 fortheinterfaceheight[22,51– consistent with respect to the resulting estimates for 𝛽. − ∝ 53]: They provide the best description of the data with only the term 𝑡 2𝛽, but one or two additional odd terms ℎ(𝑡)=sign(𝜆) (Γ𝑡)𝛽𝜒+𝜉+𝜁𝑡−𝛽 , present (𝑁∝=−2,3). The best fits resulting from mod- · els (26) are c̃︀onsistently better than those of (25), across where𝜆,Γ,𝜉 and𝜁 aremodel-dependentparametersand all datasets, which justifies discarding the latter class of 𝜒 is a universal random variable with GOE distribution modelsandtherebysupportstheKPZansatzhypothesis. in case of a flat initial condition. The KPZ ansatz hy- To obtain estimates of 𝛽 for each dataset, the best- pothesis states, that a generalisation of this form should fit value was used. The 𝛽 spread values for higher 𝑁, also hold in higher dimensions [47, 48], leading to the provide an estimate for over fitting and may serve as an following ansatz for the effective growth exponent of the error estimate for a small confidence interval of 1𝜎. For height: simulationswithRSdynamics,thisyields𝛽 =0.2414(2), which is remarkably identical to the result based on an 𝑁 𝛽 =𝛽+ 𝑐 𝑡 𝑛𝛽 , (25) average of the late-time plateau of 𝛽eff. eff,ℎ 𝑛 − The estimates obtained this way may be less reliable 𝑛=1 ∑︁ for SCA datasets, because here the model can only fit a with non-universal parameters 𝑐 and 𝑁. Higher mo- fraction of the available time series which must exclude 𝑛 ments of the height ℎ𝑛 show corrections 𝑡−𝑛𝛽, ac- theinitialoscillations. Itcanbenoted,thatthe𝛽𝑝=0.5 = cordingly, and thus ⟨ 𝑡⟩ 2𝛽 for the roughnes∝s, prescrib- 0.2411(2)isbasicallythesameastheoneobtainedbefore. − ing: ∝ For large 𝑝 the estimate 𝛽𝑝=0.95 =0.2409(4) agrees with thepreviousoneaswellas,marginally,withtheonefrom 𝑁 RS runs. However, this result is obtained because, the 𝛽 =𝛽+ 𝑐 𝑡 2𝑛𝛽 (26) model predicts the effective exponents to move upwards eff 𝑛 − 𝑛=1 for𝑡 . Thispredictionshouldberegardedwithcare, ∑︁ →∞ sinceitremainsunclearhowfartheKPZansatzcouldbe In the 2+1 dimensional restricted solid-on-solid model (RSOS) model, the dominant corrections to the rough- app̃︀lied for 𝑝 → 1. For 𝑝 = 0.75 we refrain from making anestimate,becausethemodeldoesnotseemtodescribe nessgrowthwerefoundtobeoforder 𝑡 4𝛽 [48],which ∝ − the data well. motivates the inclusion of higher orders in these forms. Ideally,suchamodelwouldfitthedatawellassoonasall relevant orders are included. Adding more terms should 3. The KPZ dynamical exponent not improve the fit quality further. However, with noisy data,addingmorefreeparametersinthisway,canresult The dynamical exponent 𝑧 = 𝛼/𝛽 of the KPZ class in over fitting, if not convergence-problems. is related to the roughness exponent 𝛼 by the Galilean Fig.4showsfitsresultsusingequation(26)onthepre- symmetry [12]: viously introduced datasets. It is immediately apparent, that (26) with 𝑁 = 1 does not describe the presented 2=𝛼+𝑧 =𝛼(1+1/𝛽) . (27) data, the 𝑛=2-term is required, as in case of the RSOS model. Inserting the our estimate 𝛽 = 0.2414(2) into this equa- In case of RS simulations, the ansatz appears to fit tion yields 𝛼 = 0.3889(3) and 𝑧 = 1.611(3). The lat- reasonably well early times: 𝑡 100 as well. The SCA ter is used to calculate autocorrelation exponents in the ≥ ̃︀ 7 0.242 DTrDB, TC=1,1 SCA, 𝑝=0.5 𝛽1 =0.2421(1) 𝛽1 =0.2416(1) 0.241 𝛽2 =0.2414(1) 𝛽2 =0.2411(1) 𝛽3 =0.2412(1) 0.240 𝛽3 =0.2411(1) ff 0.240 ff e e 𝛽 0.2420 𝛽 0.2415 0.2415 0.239 0.238 0.2410 0.2410 0.2405 (a) (b) 0.238 0.2405 5 4 3 5 4 3 − − − − − − 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1/𝑡 10 2 1/𝑡 10 2 − − · · ̃︀ ̃︀ SCA, 𝑝=0.75 SCA, 𝑝=0.95 𝛽1 =0.2415(1) 𝛽1 =0.2412(1) 𝛽2 =0.2410(1) 0.240 𝛽2 =0.2409(1) 0.240 𝛽3 =0.2409(1) 𝛽3 =0.2408(1) ff ff e e 𝛽 0.2415 𝛽 0.238 0.2410 0.238 0.2410 0.2405 0.2400 0.2405 (c) 0.236 0.2395 (d) 0.2400 0.236 5 4 3 5 4 3 − − − − − − 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1/𝑡 10 2 1/𝑡 10 2 − − · · FIG. 4: Effective exponents 𝛽 for roughness growth with KPZ ansatz fits using the form (26) to orders one through three. eff ̃︀ ̃︀ Theresultingasymptoticvaluesfor𝛽 aregiveninthelegendsaccompaniedbytheuncertaintyofthefitparameter. Theinsets show a zoom to the late-time region 1×10−5 ≤ 1/̃︀𝑡 ≤ 3×10−3 (truncated before the finite size break down Fig. 2(b)). The 1/̃︀𝑡 axes are logarithmically scaled using base 10, the labels only show the exponents. Panel (a) shows the RS dataset using DTrDB with TC=1,1. Fits were performed in the interval 1×10−5 ≤1/̃︀𝑡≤1×10−2. Panels (b)-(d) show SCA datasets with 𝑝 = 0.5,0.75 and 0.95, respectively. The fits were restricted to the interval shown in the inset. The sample size for DTrDB, TC=1,1 was 𝑛≥1044. See the captions of Fig. 2 for other sample sizes. next section. It should be noted, that the above value B. Autocorrelation for 𝛼, while in agreement with earlier numerical estimates [37, 46, 54, 55], marginally disagrees with the cur- 1. Autocorrelation of interface heights in the KPZ case rently most accepted one 𝛼=0.3869(4) [56]. Combining this roughness exponent results with our own estimate for 𝛽 violates equation (27) by about 2.5𝜎. This may a. Aging Theautocorrelationresultsoftheinterface be marginal, but still, we use our 𝛼 and 𝑧 estimates for heights under RS dynamics are summarized in Figure 5. consistency. A near-perfect collapse of the 𝐶 (𝑡,𝑠) functions could ℎ be achieved by using 𝑏 from the relation (14) (see Fig- ure 5(a)). However, the best collapse was obtained with the following aging exponent: 𝑏 = 0.469(3). This is h − the consequence of strong corrections to scaling at early times, where 𝛽 (𝑡) is smaller than the asymptotic value eff 8 TABLEI: Summaryoffittingof𝛽 usingtheKPZansatzinthemostgeneralform(25)anduing(26)withevenpowers,more eff suitable to describe corrections of the roughness-scaling up to maximum orders 𝑁 =6. Error margins given are uncertainties ofthefitparameters,notactualerrorestimatesfor𝛽. Thereducedsumsofresiduals𝜒 aregiventojudgethequalityofthe red fits. For each dataset we underlined the value of 𝜒 , where the quality does not improve significantly by increasing 𝑁. red RS SCA DTrDB, TC=1,1 DTrDB, TC=2,2 𝑝=0.5 𝑝=0.75 𝑝=0.95 𝑁 𝛽 𝜒 𝛽 𝜒 𝛽 𝜒 𝛽 𝜒 𝛽 𝜒 red red red red red equation (25) 1 0.2430(1) 10.04 0.2433(1) 9.24 0.2420(1) 7.03 0.2419(1) 8.63 0.2418(1) 3.15 2 0.2396(1) 1.82 0.2396(1) 1.29 0.2403(1) 1.62 0.2400(1) 1.31 0.2403(1) 0.51 3 0.2411(1) 0.79 0.2408(1) 0.55 0.2414(1) 0.64 0.2404(1) 1.18 0.2401(1) 0.49 4 0.2421(2) 0.67 0.2421(1) 0.32 0.2414(1) 0.64 0.2385(1) 0.63 0.2403(2) 0.48 5 0.2386(3) 0.47 0.2409(2) 0.28 0.2388(3) 0.51 0.2394(3) 0.61 0.2399(5) 0.48 6 0.2377(8) 0.47 0.2403(5) 0.28 0.2333(9) 0.45 0.2420(7) 0.59 0.2485(8) 0.40 equation (26) 1 0.2421(1) 7.28 0.2422(1) 6.27 0.2416(1) 5.22 0.2415(1) 6.54 0.2412(1) 1.76 2 0.2414(1) 1.16 0.2414(1) 1.10 0.2411(1) 0.65 0.2410(1) 1.72 0.2409(1) 0.81 3 0.2412(1) 0.65 0.2412(1) 0.33 0.2411(1) 0.64 0.2409(1) 1.51 0.2408(1) 0.52 4 0.2413(1) 0.53 0.2412(1) 0.29 0.2412(1) 0.59 0.2407(1) 0.86 0.2407(1) 0.50 5 0.2412(1) 0.51 0.2412(1) 0.28 0.2412(1) 0.59 0.2406(1) 0.68 0.2405(1) 0.41 6 0.2412(1) 0.51 0.2412(1) 0.28 0.2411(1) 0.57 0.2403(1) 0.50 0.2407(1) 0.38 (see Figure 2(b)). simulations using DTrDB, TC=1,1; the above estimates b. Autocorrelation exponent obtained by RS dynam- are also compatible with results obtained using TC=2,2; ics We calculated effective exponents of 𝜆 /𝑧 and its as evidenced by figure 5(c). 𝐶,ℎ 𝑡/𝑠 behavior, by an analysis shown on Figure 5(b). c. SCA autocorrelation functions and aging SCA →∞ In order to read-off the appropriate correction to scal- updatesarespatiallycorrelated,thereforetheyintroduce ing we linearised the left tail of the curves by plotting acontributiontotheautocorrelationfunction, whichde- themonthe 𝑠/𝑡scale. Thisleadstotheextrapolations pends on the update probability 𝑝 < 1. If we want to 𝜆 /𝑧 = 1.254(9) depending on 𝑠. The aging behavior modelcellularautomatonlikesystemsthisisnotaprob- 𝐶,h √︀ should be independent of 𝑠, but corrections arising at fi- lem,butfordescribingtheKPZequationthisisartificial. nite 𝑠 values can affect the extrapolation results. This is Figure6comparestheautocorrelationfunctionsofheight demonstrated on Figure 5(c). The plot suggests a rough variables at 𝑝 = 0.95 and 𝑝 = 0.5. The most appar- asymptotic trend of decreasing effective exponents, but ent property is the finite asymptotic value (Figure 6(a)). does not allow one to make a clean fit. This is the consequence of frozen regions, arising in or- To clarify the situation we attempted a different type dered domains, which are difficult to randomize by the oflocalslopeanalysispresentedinFigure5(d),usingtail SCA dynamics. In the dimer model updates can happen effective exponents, where each 𝜆 /𝑧 value was deter- at the boundaries only, besides this alternating domains 𝐶,h mined as the exponent of a PL-fit to 𝐶 (𝑡,𝑠) for 𝑡 𝑡. are also stable in case of SCA, they flip-flop at even-odd ℎ ′ ′ ≥ These can be expected to converge more monotonically sublattice steps, when 𝑝 1. → to the asymptotic value as before, because the left tail We applied an iterative fitting procedure to determine data of 𝐶 (𝑡,𝑠) are included in the procedure for all 𝑡 the functional behavior as follows. As a first approxima- ℎ min with an increasing weight as 𝑡 increases. Indeed, the tion the 𝐶 (𝑡 ,𝑠,𝑝)=𝑜(𝑝) limit was determined us- min h →∞ curves of different 𝑠 values in Figure 5(d) behave more ing a linear extrapolation from the function’s right tail. linearly with some additional oscillations. However, all Subtracting the appropriate value from each curve re- curves seem to fluctuate around a common mean, which vealedaPLapproachtothisconstant. Toobtainrefined is not the case in Figure 5(b). A single linear fit for the 𝑜(𝑝) values, the exponent 𝑥 was read off from the data, combination of all curves, yields an averaged extrapola- allowing a subsequent fit for the tail in the form: tion of 𝜆𝐶,h/𝑧 = 1.23(3) in a marginal agreement with 𝑓(𝑡)=𝑜+𝑐 𝑡 𝑥, (28) our previous result for this 𝜆𝐶,ℎ/𝑧 = 1.21(1) [39]. The · − present̃︀larger error margin takes the uncertainty due to with free parameters 𝑜 and 𝑐. The corrected exponents the actually unknown corrections into account. Using converged as 𝑥 𝜆 /𝑧, after subtracting the refined ′ 𝐶 → our 𝑧 value the corresponding autocorrelation exponent 𝑜(𝑝) values. These iterations yielded self-consistent esti- is 𝜆 = 1.98(5). Figure 5 mostly shows data for mates for 𝑜(𝑝) and the autocorrelation exponent of the 𝐶,heights 9 1.4 𝑠=30 𝜆𝐶,h,𝑠=30/𝑧 =1.2630(6) 8 10−1 𝑠=100 𝜆𝐶,h,𝑠=100/𝑧 =1.2593(5) 2 1.3 48 𝑠=500 𝜆𝐶,h,𝑠=500/𝑧 =1.2418(9) 0.− 10 2 𝑠=1000 /𝑧 𝜆𝐶,h,𝑠=1000/𝑧 =1.2536(13) 𝑠 − ff e 1.2 · h, ) , 𝑠 10 3 𝐶 , − 𝜆 𝑡 ( 1.1 h 𝐶 10−4 (a) (b) 1 100 101 102 103 0 0.2 0.4 𝑡/𝑠 𝑠/𝑡 √︀ 1.25 𝑠=30 (c) 𝑠=100 1.26 𝑠=500 𝑧 𝑠=1000 / 𝑧 / ff 𝜆𝐶,h/𝑧 =1.23(3) h ,e 1.2 , h 𝜆𝐶 1.25 𝐶, 𝜆 TC=2,2 (d) 1.15 TC=1,1 1.24 0 1 2 3 0 5 10 2 0.1 0.15 0.2 − · 1/𝑠 10−2 𝑠/𝑡min · FIG. 5: Autocorrelation results from RS calculations using DTrDB with TC=1,1. System size 𝐿=216, 𝑛≥1044 realizations for 𝑠 > 30 and 𝑛 ≥ 473 for 𝑠 = 30. (a) Collapsed autocorrelation functions for waiting times 𝑠 = 30,100,500,1000. (b) √︀ Correspondinglocalslopeanalysisandextrapolationsassumingcorrectionsoftheform 𝑠/𝑡,asdrawn. Linearfitwasperformed √︀ for 𝑠/𝑡∈[0.1,0.3]. Statederrorsarepurefit-errors,seetextforactualerrormargins. (c)Exponents𝜆 /𝑧(𝑠)asobtainedin 𝐶,h panel(b),correspondingvaluesobtainedinDTrDBTC=2,2simulationsaredisplayedadditionally. (d)Taileffectiveexponents corresponding to panel (a) obtained from PL fits for intervals 𝑡 ≥ 𝑡 with successively increasing 𝑡 . A linear fit to the min min combination of all curves is displayed as a solid black line. SCA. This procedure is more prone to statistical error the corrected SCA and the displayed RS autocorrelation for small 𝑡/𝑠, because 𝐶 (𝑡,𝑠) is farther away from the functions show identical behavior. h asymptotic behavior in this case, allowing noise in the tail to influence the extrapolated value more strongly. d. Autocorrelationexponent: SCA Localslopeanal- Table II lists the calculated 𝑜(𝑝) limits (including those yses of the corrected autocorrelation functions are dis- for the lattice-gas variables, see Section IIIB3). played in figures 6(c) and 6(d) at 𝑝 = 0.95 and 𝑝 = 0.5, Thelimitingvalueturnedouttodependexponentially respectively. Assumingarescalingoftheabscissa: 𝑠/𝑡, on 𝑝. Note, that similar 𝑒𝑝 dependence has been found allowsonetoobservealinearbehavioroftheeffectiveex- √︀ in 𝑓 (𝑡,𝑠) relating SCA and RS timescales [67]. ponents for intermediate times. In case of 𝑝 = 0.95 the SCA Figure 6(b) shows the corrected 𝐶 (𝑡,𝑠) functions, af- 𝑜-values could not be determined precisely enough for h tersubtractingthelimiting𝑜(𝑝)values. Anearlyperfect 𝑠>100,thusweconsideredextrapolationsat𝑠=30,100 data collapse could be achieved using the aging expo- only in a weighted average of the results. This yielded: nent 𝑏 , coming from the RS simulations. Even more, 𝜆 /𝑧 = 1.26(1) and so 𝜆 = 2.01(2). These values h 𝐶,h 𝐶,h 10 101 𝑠=30 28 10−1 𝑠=30 8 𝑠=100 4 𝑠=100 𝑠=500 0.− 10−2 𝑠=500 100 𝑠=1000 𝑠 𝑠=1000 ) 𝑡,𝑠 𝑜)·10−3 D𝑠T=rD10B0, (h 10−1 𝑝 = 0.95 − 𝐶 10 4 ) − 𝑠 , 10−2 (a) 𝑝 = 0.5 (𝑡h 10−5 (b) 𝐶 ( 10 6 − 100 101 102 103 104 105 100 101 102 103 104 𝑡/𝑠 𝑡/𝑠 1.4 1.4 𝜆𝐶,h,𝑠=30/𝑧 =1.2591(7) 𝜆𝐶,h,𝑠=30/𝑧 =1.2582(6) 𝜆𝐶,h,𝑠=100/𝑧 =1.2613(6) 𝜆𝐶,h,𝑠=100/𝑧 =1.2404(12) 1.3 1.3 𝜆𝐶,h,𝑠=500/𝑧 =1.2430(8) 𝜆𝐶,h,𝑠=500/𝑧 =1.2375(8) /𝑧 𝜆𝐶,h,𝑠=1000/𝑧 =1.2434(9) /𝑧 𝜆𝐶,h,𝑠=1000/𝑧 =1.2408(4) ff ff e 1.2 e 1.2 , , h h , , 𝐶 𝐶 𝜆 𝜆 1.1 1.1 (c) (d) 1 1 0 0.2 0.4 0 0.2 0.4 𝑠/𝑡 𝑠/𝑡 √︀ √︀ FIG.6: AutocorrelationofKPZheightsfromSCAcalculations. Errorbarshavebeenomittedforclarity. Thevisiblenoiseis agoodindicationfor1𝜎 error. Panels(a)and(b)showdatasetswith𝑝=0.5(3062realizations,𝑡≤1.4MMCS)and𝑝=0.95 (3062 realizations, 𝑡 ≤ 400kMCS), curves from bottom to top. Lateral system size is 𝐿 = 216. (a): Raw autocorrelation functions showing saturation depending on 𝑝. (b): Collapsed autocorrelation functions, corrected by the saturation offset 𝑜 (see text). Plots for 𝑝 = 0.5 and 𝑝 = 0.95 corresponding to the same 𝑠 use the same colors, where the bottom set of plots corresponds to 𝑝 = 0.5. Colors are less saturated for the plots for 𝑝 = 0.5, to distinguish them at late times. The DTrDB, TC=1,1 autocorrelation function of 𝑠 = 100 is also displayed for comparison. The bottom panels (c) and (d), show the local slopeanalysiscorrespondingtothe𝑝=0.95and𝑝=0.5datasets,respectively. Extrapolationsassumecorrectionsoftheform √︀ 𝑠/𝑡, as drawn. Printed error margins are pure fit-errors. are in good agreement with those obtained from a local 2. Autocorrelation of interface heights in the EW case slope analysis of RS calculations for small 𝑠. Since the autocorrelation function in the EW case is The effective exponents for 𝑝 = 0.5 show a slightly known exactly (22), we can verify our simulations by a decreasing tendency with 𝑠 in Figure 6(d), moving to- comparison with it. Indeed, the expected form could be wards the RS estimate 𝜆 /𝑧 = 1.23(3). However, we reproduced by our RS implementation. A more interest- 𝐶,h can’t consider the extrapolated values for 𝑠 = 500 and ingresultis,thattheSCAsimulationsalsofititperfectly. 𝑠=1000moreprecise,th̃︀anthoseat𝑠=30,100,because Thefinitesaturationvalue,causedbycorrelatedupdates, thedeterminationofthe𝑜(𝑠)constantbecomesmoreun- observed in the KPZ case is not present here. certain at higher times, increasing the possible error of The agreement with the analytical form is exemplified the exponent estimates. inFigure7(b). Asmalldeviationatveryearlytimescan