Department of Physics

Seminar I - 1st year, II. cycle

Stochastic dynamics of surfaces andinterfaces

Author: Timotej LemutAdvisor: assoc. prof. Marko Znidaric

Ljubljana, 2018

Abstract

In the seminar, we are interested in the behaviour of driven stochastic surfaces. We introduceroughness as standard deviation of height of the surface averaged over noise and study its large scalebehaviour. From a systematical treatment of a general Langevin equation we arrive at two equationsfor height, namely Edward-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) equation. They bothrepresent each its own universality class. We state some properties of both equations. We derivecritical exponents of the EW class in any dimension and get an explicit equation for roughness. ForKPZ we are able to find out the critical exponents only in dimension d = 1.

Contents

1 Introduction 1

2 Introduction of roughness and its critical exponents 2

3 Universality classes of random deposition 3

4 Edward-Wilkinson equation 4

5 Kardar-Parisi-Zhang equation 6

6 Discussion 8

1 Introduction

We will be interested in behaviour of surfaces in the presence of noise, that are also driven in some way.Surface is described with a height function and is usually realised as some sort of boundary betweentwo phases. Examples could be purely mathematical models or some table-top experiments, for exampleburning or wetting front of a paper. Other experimental realisations include boundaries between solidand liquid/gaseous phases of matter, domain walls, and boundaries between different phases in liquidcrystals. For example, in the case of a boundary between solid and gaseous phase, driving is representedby deposition or evaporation of particles, while noise would be represented by thermal fluctuations.

One of the simplest models of random interfaces are deposition models. Below we describe some ofthose, to which we will reference later in the seminar and which will help to illustrate some notions.They belong to different universality classes, meaning, have different large scale behaviour.

Figure 1: Schemas of two models of random interfaces: A - Random deposition model with relaxation,B - Ballistic deposition [1].

The most straightforward is called the random deposition model (RD). It is defined over integersand its evolution rule is: pick a random site and deposit a particle on it. Height is the number ofparticles already deposited on each site. Adding a rule, that after every deposition on a randomly chosensite, we look at the neighbouring sites of the newly deposited particle and, if possible, we move theparticle to one of the sites that are lower than the site on which the particle has been deposited, we getto the model called random deposition with relaxation model (RDR) and one realisation with severalpossible depositions of a new particle is depicted in figure 1.A. Simple variation of RD would be not torelax a particle after deposition, but apriori restrict the difference in height of each neighbours, so whena deposition of a particle on a chosen site would violate a restriction of maximum height difference, aparticle would not be deposited. Such model is called restricted solid on solid model (RSOS) and actuallybelongs to a different universality class than RDR.

In figure 1.B a picture of falling particles in the ballistic deposition model (BD) is displayed, where theparticle is again deposited to a randomly chosen site, but sticks to the particles of the neighbouring sites.As we can see, voids and overhangs can form. We also include a picture (Figure 2) of some realisationsof all the described models.

In the second chapter we introduce roughness of the surface as a standard deviation of height of thesurface averaged over noise. We guess that it diverges in the limit of infinite samples and infinite time,so we introduce its scaling form and define the critical exponents describing its divergence. In the next

1

Figure 2: Realisations of models: A - RD, B - RDR, C - BD, D - RSOS [2]. We notice a much roughersurface in the case of RD, where the sites are completely independent of each other. Voids and overhangsare visible in the case of BD model.

chapter we derive simplest equations that could govern stochastic boundaries. We only use argumentsof symmetry and power counting. In the next two chapters, we examine two of such equations. The firstone is called Edward-Wilkinson (EW, [3]) equation and is actually just diffusion equation with additivenoise, while the other one, called the Kardar-Parisi-Zhang (KPZ, [4]) equation, is nonlinear. We are ableto get some intuition about its solutions and even determine critical exponents in dimension d = 1. ForEW we are able to do that in arbitrary dimension and even get to the explicit expression for roughness.

2 Introduction of roughness and its critical exponents

We imagine a boundary as a surface (for easier analysis without overhangs) defined over volume V inarbitrary dimension d, which we will represent with height function h(x, t), where x is d-dimensionalvector and t is time. We define roughness of the surface W (L, t) as a function of its linear size L = V 1/d

and time t asW 2(L, t) =

⟨h2(x, t)− h(x, t)

2⟩

, (1)

where the overline · denotes the spatial average and 〈·〉 the average over noise (averaging over realisations).Looking back to the introduced models, for RD, we can already guess how its roughness will behave

with respect to L and t. We notice that there is no coupling between different sites, which means thatthe size and the spatial dimension of the system L are irrelevant. RD is just a collection of independentrandom walks, where W does not depend on L at all, while dependence of W on time is the usualW (L, t) ∝

√t.

For roughness of finite sample not to diverge with time, there has to be some sort of smoothingmechanism, some interaction between neighbouring points on the interface, so that the finite surfacecannot get arbitrarily rough. All other above models have some kind of coupling between sites, soroughness saturates after some critical time and also depends on L,

limt→∞

W (L, t) = W (L, t∞). (2)

For those cases, we expect that by taking an infinite sample, we will nevertheless encounter arbitrarilylarge deviations in height,

limL,t→∞

W (L, t) =∞. (3)

By making that observation, we go on the same route as that of equilibrium phase transitions, byhaving free energy density replaced with roughness. On the grounds of the above considerations, one

2

uses the scaling

W (L, t) = Lαw

(t

Lz

), where w(u) ≈

{uβ , u� 1

const., u� 1, (4)

and α is called the roughness exponent, β is the growth exponent and z = α/β the dynamic exponent.By using above ansatz we also calculate what kind of scaling transformation produces statistically

the same profile. We are checking whether the interface we are describing is possibly self-similar (heightprofile scales with the same factor as space), or something similar.

If we rescale space by a factor of b, time must be scaled by bz, so that the argument of the function win equation (4) stays the same. Also from the same equation, we have that roughness will scale with bα

and the same holds for height also, since it is linear in roughness. Scaling transformations are therefore:

x→ bx

t→ bzt (5)

h→ bαh.

Interface which scales in such a way is called self-affine.In the introduction, we also consider dependence of critical exponents on dimension. There exist

lower and upper critical dimension, d`c and duc , which are defined as:

α(d`c) = 1 and α(duc ) = β(duc ) = 0. (6)

Meaning, for d > duc , fluctuations do not roughen the surface on large time and length scales, while ford < d`c, fluctuations make the surface super-rough. In that case not only does the difference of heightdiverge but the slope also diverges.

3 Universality classes of random deposition

We examine a general Langevin-type equation for a moving boundary

· · ·+ ∂2t h(x, t) + ∂th(x, t) = A[h] + η(x, t), (7)

where A is some functional of h and η is white noise, meaning

〈η(x, t)〉 = 0 (8)

〈η(x, t)η(x′, t′)〉 = Γδ(x− x′)δ(t− t′). (9)

Now we use some sound assumptions to specify A a bit more precisely. First, we assume that A doesnot depend explicitly on x and t, which would be violated by some time(or space)-dependent driving,for example. Other assumption is that the system is also spatially invariant in the growth direction,meaning under transformation h(x, t) → h(x, t) + h0. If our two conditions are satisfied, our equationlooks like

· · ·+ ∂2t h(x, t) + ∂th(x, t) = A[∂ih, ∂ijh, . . . ] + η(x, t) (10)

To get even more precise with the form of A, we can check which of its terms are the most relevanton large scales (which is implemented by taking transformation (5) and sending b to infinity). First ofall we can get rid of all higher order time derivatives since a term ∂nt h gains a factor b−nz by scalingtransformation, so only the first time derivative survives. Now for the functional A, which can be formallyexpanded as

A = A0 +

d∑i=1

Ai∂ih+

d∑i,j=1

Aij∂ih∂jh+

d∑i,j=1

Bij∂ijh+ . . . , (11)

we first find out that we can cross out the constant term A0, by defining new height h′(x, t) = h(x, t)−A0tand also the linear term by taking h′(x, t) = h(x−At, t), so that we are left with

A =

d∑i,j=1

Aij∂ih∂jh+

d∑i,j=1

Bij∂ijh+

d∑i,j,k=1

Aijk∂ih∂jh∂kh+ . . . , (12)

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which we write in condensed notation as

A = {2, 2}+ {1, 2}+ {3, 3}+ . . . . (13)

The first number in a pair {p, q} represents a number of times h appears and q is the number of derivativesin each term. We emphasize, that from our previous arguments about spatial invariance in the growthdirection, we must have p ≤ q, otherwise an explicit dependence on h appears. Under the transformation(5) we have

{p, q} → bpα−q{p, q}. (14)

Since p ≤ q, the leading term is {p, p}, which transforms as

{p, p} → bp(α−1){p, p}. (15)

So, for α < 1 or with other words for d`c < d < duc , we expect the most relevant term of A to be {2, 2},since we crossed out the terms {0, 0} and {1, 1}. Matrix Aij can be symmetrized and consequentlydiagonalized. By rescaling every xi by a suitable factor and demanding rotational invariance, we are leftwith a term proportional to (∇h)2.

Such an equation is invariant under xi → −xi, but not under h → −h. If we demand for both ofthese two symmetries to hold, then q must be even and p odd, meaning that the leading term is now{1, 2}, which by the same procedure as before becomes a term proportional to ∇2h.

By such general consideration of symmetries and power counting, we arrive at two equations whichare called Edward-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) equations, respectively:

∂th(x, t) = ν∇2h+ η(x, t) (16)

∂th(x, t) = ν∇2h+λ

2(∇h)2 + η(x, t), (17)

where the diffusive term is also included in the KPZ, to regularize the behaviour at small scales, eventhough compared to the term proportional to (∇h)2 it is irrelevant at large scales. We also stress thatthe EW equation is linear, whereas the KPZ equation is not.

4 Edward-Wilkinson equation

In this section we examine in some detail the so-called EW equation (16), where ν > 0 and η obeys (8)and (9).

First we notice, that the equation can be written in the conserved form

∂th(x, t) = −∇ · J + η(x, t), where J = −ν∇h, (18)

meaning the average height is a conserved quantity, as can be easily shown

dh

dt=

1

V

∫V

η(x, t)dx = 0, (19)

since the integral of the divergence vanishes for periodic boundary conditions. As we can see, we havea simple interplay between noise and diffusion, one roughening and the other one smooting the surfacewith the current J.

From the above introduced models only the RDR belongs to the EW universality class, meaning ithas the same large scale behaviour as a solution of the EW equation. If we imagine that height in theRDR actually represents number of particles on each site, then the RDR corresponds to a diffusion-likeprocess with the diffusion constant ν in the presence of uniform supply of particles.

We also notice, that the equation can be written with a potential in the form

∂th = −δFδh

+ η, where F [h] =ν

2

∫V

(∇h)2dx, (20)

which implies, that the dynamics evolves in order to minimize that potential, as we can compute readily:

dFdt

=

∫V

δFδh

δh

δtdx (21)

= −∫V

(δFδh

)2

dx +

∫V

δFδhη(x, t)dx, (22)

4

so that ⟨dFdt

⟩≤ 0. (23)

The meaning of F is also at hand, by remembering that the total area S of surface h(x) is

S =

∫V

√1 + (∇h)2dx = V +

1

2

∫V

(∇h)2dx +O((∇h)4

). (24)

Therefore, neglecting terms irrelevant for large scales we get F = νS and the minimization of potentialis just the minimization of total surface area. A consequence of having a potential is that the propertiesof the stationary state (so when t � Lz, equation (4)) are equivalent to an equilibrium state describedby the Hamiltionian F .

We now first show that the critical exponents of EW universality class can be obtained by a simpleconsideration of units and after that we also indicate how to get an explicit equation for roughness byFourier transform of the EW equation.

First, for units of length, time and height we write, respectively, [L] = L, [t] =T and [h] =H.1 We nowwrite equation (4), including all posible dependence on Γ and ν,

W (L, t) = Γa1νb1Lαw

(Γa2νb2

t

Lz

)(25)

and impose that the left and right hand side of the equation have the same units, meaning the argumentof the function w is dimensionless and [W ] =H. We get the expressions for [Γ] and [ν] from equations(16) and (9), respectively:

H

T=

[ν]H

L2= [η] (26)

[η]2 =[Γ]

LdT. (27)

Using above relations in the expression for W (L, t) we get two equations(LdH2

T

)a1 (L2

T

)b1Lα = H, (28)(

LdH2

T

)a2 (L2

T

)b2T = Lz, (29)

from which we finally get for the critical exponents of the EW universality class

α =2− d

2, β =

2− d4

, z = 2. (30)

From (6) we can also get lower and upper critical dimensions for EW class to be 0 and 2, respectively.We can also get an explicit form of w, by Fourier transforming the EW equation. Writing

h(x, t) =1

(2π)d

∫eiq·xh(q, t)dq, (31)

we get the solution of the EW equation to be

h(q, t) = h(q, 0)e−νq2t +

∫ t

0

e−νq2(t−t′)η(q, t′)dt′. (32)

For roughness, we will need just the correlator 〈h(q, t)h(q′, t)〉, since h(x, t) = 0. We can also change theorder of evaluating averages and lose the spatial average, since we know that the system is translationallyinvariant and the spatial dependence will therefore disappear after averaging over noise:

W 2EW (L, t) =

⟨h2(x, t)

⟩= 〈h2(x, t)〉 =

⟨h2(x, t)

⟩=

1

(2π)2d

∫ei(q+q′)·x 〈h(q, t)h(q′, t)〉 dqdq′. (33)

1We choose separate unit for height, since it may represent some other quantity, not necessary measured in length units.Also, EW equation describes the dynamics of self-affine interface, not necessarily self-similar, as we have shown above,meaning that the two behave differently and could be measured in different units.

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We obtain the correlator from the solution of EW equation (32) by noticing that the first term of thesoution will vanish for initially flat surface, or after some transient time. Correlator of height is thereforecalculated from the correlator of noise, which we know from the definition. Putting everything togetherwe get for roughness

W 2EW (L, t) =

Γ

(2π)d

∫Vq

1− e−2νq2t

2νq2dq, (34)

where we are integrating over Vq = {q | πL ≤ |qi| ≤πa0}, a0 being a cutoff for avoiding integration over

arbitrarily small wavelengths, and is justified by the ultimate discrete nature of matter/our model.We can check that the above expression matches with formulas (4) and (30). By a change of variables

s = Lq, we get

W 2(L, t) =

[√ΓL2−d

2νw

(νt

L2

)]2, with w2(u) =

1

(2π)d

∫Vs

1− e−2s2u

s2ds, (35)

where exponents α and z really are the same. To check for β, we make another change of variablesy = u1/2s, so that

w2(u) =u

2−d2

(2π)d

∫Vy

1− e−2y2

y2dy, (36)

where for u� 1 the integral goes to a constant and w(u) ≈ u 2−d4 , satisfying the relation β = α/z.

5 Kardar-Parisi-Zhang equation

The second equation we will consider is called the KPZ equation (17), where again ν > 0 and η is noiseobeying (8), (9).

Here, we state the key differences from the EW equation: it is nonlinear, it is not symmetric toinversion h → −h and h(x, t) is not conserved. Namely, if we act with spatial average on the equationwe get

∂th =λ

2(∇h)2. (37)

The average height velocity is proportional to square of the slope, meaning that the surface obeying KPZequation will exhibit some kind of moving interface.

All of the above properties are shared with the BD and RSOS model and both of them actually dobelong to the KPZ universality class.

We can take a look at a deterministic version of the KPZ (without noise term), to see how a parabola,which turns out to be a self-similar solution of the deterministic KPZ, evolves. If we imagine an initialcondition for height as a parabola of positive/negative curvature positioned at the origin, we see thatthe nonlinear term (for λ > 0) will in both cases be positive and get bigger away from the center, whilethe linear term is positive/negative and constant. All that amounts to the positively curved parabola tomove upward and getting more curved with time, while the negatively curved parabola moves downwardand flattens.

Next, we show that the deterministic KPZ can actually be solved exactly in any dimension with theso-called Cole-Hopf transformation [5], [6], given by

H(x, t) = eλ2ν h(x,t). (38)

By inserting in the equation, we get the diffusion equation for H(x, t) with the diffusion coefficient ν,whose solution is known to be

H(x, t) =

∫1

(4πνt)d/2e−

(x−x′)24νt H(x′, 0)dx′, (39)

or in terms of height

h(x, t) =2ν

λln

[∫1

(4πνt)d/2e−

(x−x′)24νt + λ

2ν h(x′,0)dx′

]. (40)

We would imagine that the same transformation would help with solving the stochastic equation as well,but we get an additional problem of transforming additive noise to the multiplicative one:

∂tH = ν∇2H +λ

2νHη(x, t). (41)

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In mathematically not very strict language, the problem of multiplicative noise can be explained asfollows: we imagine the noise in the equation as a collection of delta functions and according to theequation, each delta function in the noise term will result in a jump of the function H, where the valueof H at the exact time the delta function arrives is undetermined. So the problem with understandingsuch an equation as above is whether to substitute for H, at the time delta arrives, the value of H beforethe jump, after the jump or perhaps the mean of both.2

Next, we notice that we get a familiar equation by taking a spatial derivative of both sides of theKPZ equation. Writing m = ∇h and applying gradient to both sides of the KPZ, we get the equation

(∂t − λ(m · ∇))m = ν∇2m +∇η (44)

which is, for u = −λm, like Navier-Stokes equation of an incompressible fluid when pressure effects andexternal forces can be neglected with extra noise term. We can easily show, that the above equation isinvariant under Galilean transformation

u′(x, t) = u(x− u0t, t) + u0. (45)

Now we check what that means for the KPZ equation. Writing the above equation in terms of height,we get after integration over the space variable x

h′(x, t) = h(x− u0t, t)−1

λu0 · x + a(t). (46)

By inserting above transformation in the KPZ equation, or in the equation (37), we get the expressionfor a(t) and so we get to the so-called tilt transformation

h′(x, t) = h(x− u0t, t)−1

λu0 · x +

u202λt, (47)

under which the KPZ equation is invariant.3

Invariance under tilt transformation can give us some more insight on the KPZ dynamics. Sincethe tilt transformation (47) explicitly depends on λ and the invariance under that transformation musthold for all length scales, we conclude, that λ cannot be renormalized by a change of scale.4 Under thetransformation (5) each term in the KPZ equation transforms as

bα−z∂th(x, t) = νbα−2∇2h+λ

2b2α−2(∇h)2 + b−

d+z2 η(x, t) (49)

which we rewrite as

∂th(x, t) = νbz−2∇2h+λ

2bα+z−2(∇h)2 + b

z−2α−d2 η(x, t). (50)

Since λ does not renormalize, we require that bα+z−2 = 1 or

α+ z = 2. (51)

The above relation holds for any d and tells us, that there is only one critical exponent left to bedetermined. In dimension d = 1, we are actually able to do that, as we show below.

2There are two popular approaches [7]. Taking the value of H before the jump (Ito):

H(t+ ∆t) = H(t) + ν∇2H(t) +λ

2νH(t)

∫ t+∆t

tη(t′)dt′ (42)

or the mean of values before and after (Stratonovich):

H(t+ ∆t) = H(t) + ν∇2H(t) +λ

2ν

H(t) +H(t+ ∆t)

2

∫ t+∆t

tη(t′)dt′ (43)

3Note that the invariance of the KPZ equation under the tilt transformation only holds for white noise.4By doing renormalization for parameters ν, λ and Γ we get the flow equation for each one of them, where the equation

for λ looks likedλ

d`= λ(α+ z − 2), (48)

and ` = ln b. So we have two fixed points, either λ = 0 (EW) or α+ z = 2.

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We write for our Langevin equation (where N is either EW or KPZ)

∂th(x, t) = N [h] + η(x, t), where (52)

〈η(x, t)〉 = 0 and 〈η(x′, t′)η(x, t)〉 = Γδ(x− x′)δ(t− t′), (53)

its associated Fokker-Planck equation for the probability P (h, t) of a certain height profile h occuring attime t

∂P

∂t=

∫δ

δh

[−NP +

Γ

2

δP

δh

]dx. (54)

By solving above equation and getting the expression for P (h, t) we could easily determine roughness,by calculating the first two moments of height.

Unfortunately, finding the general solution to our Fokker-Planck equation is not possible. Whatwe can do is try to find the time-independent solution of the Fokker-Planck equation, which describesstationary state attained asymptotically by a finite system. Such a solution would allow us to determinethe roughness exponent α. Such procedure can only be done for the EW equation, but we can show thatin d = 1, KPZ has the same stationary solution as EW.

Looking for a stationary solution Ps we are solving the equation (where N is again EW or KPZ)

δPsδh

=2

ΓNPs. (55)

For the EW equation NEW = − δFδh , as we have shown above, with F = ν2

∫(∇h)2dx, so that the

stationary solution isPEWs = e

2Γ

∫VNEW dh = e−

2ΓF . (56)

For the KPZ equation, the functional of the height is NKPZ = NEW + λ2 (∇h)2 and after inserting in (55)

we get the condition for PEWs to be a stationary solution of the KPZ equation∫

δ

δh

[(∇h)2PEW

s

]dx = 0. (57)

Taking the functional derivative, we obtain∫δ

δh

[(∇h)2PEW

s

]dx = −2PEW

2 δ(0)

∫∇2h(x)dx +

2η

ΓPEWs

∫(∇h(x))2∇2h(x)dx, (58)

where the first term vanishes because of periodic boundary conditions, while the second term is evaluatedin d = 1 and d = 2 below∫

(∇h)2∇2hdx =

{∫h2xhxxdx =

∫13 (h3x)xdx = 0, d = 1∫

(h2x + h2y)(hxx + hyy)dx 6= 0, d = 2.(59)

We find out that in d = 1, EW and KPZ share the same long-time statistics of stationary fluctuations.Therefore αKPZ = αEW = 1

2 . Since αKPZ + zKPZ = 2, we get zKPZ = 32 and consequently βKPZ = 1

3 >βEW . This means that in one dimension a surface subject to the KPZ equation roughens faster than thecorresponding EW surface, but the final stationary value scales with L in the same way.

In higher dimensions both KPZ exponents are larger than EW exponents (which vanish for d ≥ 2as we noted above). The fact that a KPZ surface tends to be more rough can be traced to the weakersmoothing mechanism of its deterministic parts. The EW smoothing process is the typical diffusionalsmoothing, which induces an exponentially fast relaxation, while the relaxation of bumps in the KPZsurface follows a power law. It is therefore reasonable to speculate that in KPZ, noise should lead to arougher surface than in EW.

6 Discussion

Below we summarize our findings. We first examined the Edward-Wilkinson equation, which is a lineardiffusion equation with additive white noise. The diffusive term is the smoothing mechanism, whichcounters the roughening effect of noise, while in the KPZ equation, we have another term which amountsto lateral growth. Critical exponents of EW universality class are

α =2− d

2, β =

2− d4

, z = 2. (60)

Where α and β vanish in dimension d ≥ 2. Critical exponents of KPZ universality class are [8], [9], [10]

8

d α β z1 1/2 1/3 2/32 0.3869(4) 0.2398 1.61313 0.3135815) 0.186 1.68654 0.2537(8) 0.1453 1.7463

where for dimension d = 1, we have stated the exact exponents, while in higher dimensions values arecomputed numerically for the RSOS model. Numbers in parentheses are standard deviation, values of zand β are determined by the relations holding for KPZ class in any dimension

z = 2− α and β = α/z. (61)

Above simulations also imply that the upper critical dimension for the KPZ class is larger than 4,

duc > 4, (62)

while the exact value of duc is still unknown, with field-theoretic approach suggesting duc = 4, numerics,as we see above, duc > 4 and real-space RG giving duc = ∞. All of these approaches agree about thephase diagram of KPZ for d < 4.

To conclude our seminar, we mention two ways to motivate further research. One being trying to solvethe KPZ equation, since it appears in many other, seemingly unconnected areas, for example study ofasymmetric simple exclusion processes or directed polymers in random environments. The other topic isa generalisation of growth models to a wider class of nonlocal models, where in contrast to our introducedmodels, the growth depends on the conformation of the whole system.

References

[1] R. Livi and P. Politi, Nonequilibrium statistical physics : a modern perspective. Cambridge universitypress, 2017.

[2] Pictures retrieved from. http://slideplayer.com/slide/7857723/. Accessed: 31.5.2018.

[3] S. F. Edwars and D. Wilkinson, “The surface statistics of a granular aggregate,” Proceedings of theRoyal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 381, no. 1780,pp. 17–31, 1982.

[4] M. Kardar, G. Parisi, and Y.-C. Zhang, “Dynamic scaling of growing interfaces,” Phys. Rev. Lett.,vol. 56, pp. 889–892, Mar 1986.

[5] J. D. Cole, “On a quasi-linear parabolic equation occuring in aerodynamics,” Quart. Appl. Math,vol. 9, pp. 225–236, 1951.

[6] E. Hopf, “The partial differential equation ut + uux = µxx,” Communications on Pure and AppliedMathematics, vol. 3, no. 3, pp. 201–230.

[7] N. G. van Kampen, Stochastic processes in physics and chemistry. North-Holland publishing com-pany, 1981.

[8] A. Pagnani and G. Parisi, “Numerical estimate of the kardar-parisi-zhang universality class in (2+1)dimensions,” Phys. Rev. E, vol. 92, p. 010101, Jul 2015.

[9] E. Marinari, A. Pagnani, and G. Parisi, “Critical exponents of the kpz equation via multi-surfacecoding numerical simulations,” Journal of Physics A: Mathematical and General, vol. 33, no. 46,p. 8181, 2000.

[10] A. Pagnani and G. Parisi, “Multisurface coding simulations of the restricted solid-on-solid model infour dimensions,” Phys. Rev. E, vol. 87, p. 010102, Jan 2013.

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