PHYSICAL REVIEW LETTERS 124, 060601 (2020)

Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula1

Yohsuke T. Fukai

1,2,*

and Kazumasa A. Takeuchi

2,3,†Nonequilibrium Physics of Living Matter RIKEN Hakubi Research Team, RIKEN Center for Biosystems Dynamics Research, 2-2-3
Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo 650-0047, Japan2
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan3
Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan(Received 26 September 2019; revised manuscript received 27 November 2019;
accepted 8 January 2020; published 10 February 2020)
We study fluctuations of interfaces in the Kardar-Parisi-Zhang (KPZ) universality class with curved
initial conditions. By simulations of a cluster growth model and experiments with liquid-crystal turbulence,
we determine the universal scaling functions that describe the height distribution and the spatial correlationof the interfaces growing outward from a ring. The scaling functions, controlled by a single dimensionlesstime parameter, show crossover from the statistical properties of the flat interfaces to those of the circularinterfaces. Moreover, employing the KPZ variational formula to describe the case of the ring initialcondition, we find that the formula, which we numerically evaluate, reproduces the numerical andexperimental results precisely without adjustable parameters. This demonstrates that precise numericalevaluation of the variational formula is possible at all, and underlines the practical importance of theformula, which is able to predict the one-point distribution of KPZ interfaces for general initial conditions.
DOI: 10.1103/PhysRevLett.124.060601

Efforts on universal behavior associated with scale invariance, which have established important concepts such as therenormalization group and the universality class, now shedlight on novel aspects of nonequilibrium fluctuations. In thisrespect, the Kardar-Parisi-Zhang (KPZ) universality class[1–4] plays a distinguished role, because of the existence ofexact solutions and experimental realizations. The KPZ classis also known to arise in a variety of problems: besides
growing interfaces and directed polymers as originallyproposed [1], it also turned out to be relevant for stochastic
particle transport, quantum integrable systems [3,4], andfluctuating hydrodynamics [5], to name but a few.
In the following, let us focus on the one-dimensionalcase, for which exact studies have been developed, andconsider growing interfaces described by the height hðx; tÞat position x ∈ R and time t ∈ R. The KPZ class describes
scale-invariant fluctuations of growing interfaces in thelong-time limit, in general situations without particularsymmetries and conservation laws. The hallmarks of the
KPZ class are the scaling laws for the fluctuation amplitude∼ tβ and the correlation length ∼ t1=z , with universal
exponents β and z that take the values β ¼ 1=3 and z ¼
3=2 for the one-dimensional case [1,2,4]. The height hðx; tÞis then generally written, for large t, ashðx; tÞ ≃ v∞ t þ ðΓtÞ1=3 χðX; tÞ;ð1Þ

where χðX; tÞ is a stochastic variable, X ≔ x=ξðtÞ denotes
the coordinate rescaled by the correlation lengthξðtÞ ≔ ð2=AÞðΓtÞ2=3 , and v∞ , Γ, and A are systemdependent parameters. The variable χðX; tÞ is expected0031-9007=20=124(6)=060601(6)to be universal, in the sense that its statistical properties do
not depend on microscopic details of the systems. The
scaling exponents of the KPZ class have been found invarious experimental systems [6], including colonies ofliving cells [7,8], combusting paper [9], and liquid-crystal
turbulence [4,10–12].
Recently, remarkable developments triggered by exact
studies [3,4] have unveiled novel aspects of the KPZ class.
A particularly important outcome is the geometry dependence, which we describe below. If an interface grows on topof a flat substrate, as usually assumed in simulations, theinterface roughens but maintains the globally flat profile. In
contrast, if an interface in a plane starts to grow from a point
nucleus, say, at x ¼ 0, it takes a circular shape with agrowing radius. Although this interface becomes flatter andflatter as the radius increases, statistical properties ofχðX; tÞ remain distinct from the flat case. Specifically,χðX; tÞ has different asymptotic behavior as follows:dχðX; tÞ ⟶dðflatÞ
A1 ðXÞ;A2 ðXÞ − X2 ; ðcircularÞ
d

ð2Þ
d

where ! denotes convergence in distribution (¼ and ≃ will
be used analogously). A1 ðXÞ and A2 ðXÞ are called theAiry1 [13,14] and Airy2 [15] processes, respectively, andare well studied analytically [16]. Due to their translational
invariance, as long as one-point properties are concerned,
Ai ðXÞ can be replaced by a single stochastic variable χ i .
Remarkably, the one-point distributions of χ 1 and χ 2 wereshown [17–20] to coincide, respectively, with the GOE and060601-1© 2020 American Physical Society

PHYSICAL REVIEW LETTERS 124, 060601 (2020)
GUE Tracy-Widom distributions [21] known from randommatrix theory [22], which describe the distribution of thelargest eigenvalue of random matrices in the Gaussianorthogonal and unitary ensembles (GOE and GUE),
respectively. This geometry dependence, as well as the
emergence of the Tracy-Widom distribution, turned out tobe experimentally relevant too, as shown by experiments on
liquid-crystal turbulence [4,10,11]. Correlation properties
were also shown to be different between the flat andcircular cases, even though the scaling exponents β andz take the same values. On the basis of those results, one
may state that the flat and circular interfaces constitutedifferent universality subclasses within the single KPZclass, characterized by different yet universal distribution
and correlation properties.
Those universality subclasses have been, however,mostly studied for a few “canonical” cases including theflat and circular ones. A natural and important question isthen what happens for more general initial conditions.
Theoretically, the KPZ fixed-point variational formula[16,23–26] can be used to predict the asymptotic propertiesof χðX; t → ∞Þ for general initial conditions. On the other
hand, experimental and numerical studies have focused on
finite-time behavior emerging from intermediate initialconditions. For example, the present authors [12] studied
growth from a ring of finite radius R0 , which then produces
two curved interfaces, one growing outward and the otherone inward. Focusing on the ingrowing interfaces, we foundthat finite-time properties of χðX; tÞ for different R0 arecontrolled solely by the rescaled time τ ≔ v∞ t=R0 , asfollows: statistical properties of χðX; tÞ agree with thosefor the flat subclass initially (τ ≪ 1), until the interfaces
nearly collapse at τ ≈ 1 and therefore do not behave as KPZ
anymore. Analogous behavior was also observed numerically by Carrasco and Oliveira [27], who used lattice models
with system size set to decrease in time (mimicking theshrinking circumference of the ingrowing interfaces).
The case of enlarging substrates, which would correspondto the outgrowing case, has also been studied, and crossoverfrom the flat to circular subclasses was suggested in this case[27–29], which is also expected to be described by τ.
However, it remains unclear how universal such finite-time
behavior is, why τ is the right parameter to describe it, andabove all, how such crossover can be described theoretically.
Those problems are addressed and answered in thisLetter. We study outgrowing interfaces from ring initialconditions both numerically and experimentally, using an
off-lattice version of the Eden model [30] and the liquidcrystal turbulence [4,10–12]. Scaling functions for theflat-to-circular crossover are determined and shown to be
the same for both of the studied systems. Moreover, we
describe this crossover theoretically, by adapting thevariational formula [16,23–26] for curved initial conditions. The formula is numerically evaluated and shown toreproduce our numerical and experimental results

quantitatively, without adjustable parameters. This also
implies that the flat-to-circular crossover is indeed universal
and, furthermore, should generally appear for any curved
interfaces with locally parabolic initial conditions.
We first study the off-lattice Eden model [30], in which acluster of round particles (with unit diameter) grows bystochastic addition of new particles. The initial condition is
set to be a ring of N particles [Fig. 1(a)]. The evolution rule is
as follows (see Ref. [30] for details): At each time step, we
randomly choose a particle at the interface, attempt to put anew particle next to it in a random direction, and do so if
there is no overlapping particle. Time is then increased by
1=ðthe number of the interfacial particlesÞ whether the newparticle was added or not. Particles that cannot contribute
further growth are checked and removed from the list of theinterfacial particles with every time unit. To characterize theheight fluctuations, we measure the local radius incrementRðθ; tÞ, which is the radial distance between the initial ringand the interface at each angular position θ [Fig. 1(a)].
Thanks to the rotational symmetry, we havedRðθ; tÞ ¼ hð0; tÞ ≃ v∞ t þ ðΓtÞ1=3 χð0; tÞ;ð3Þ

but statistical precision can be improved by averaging over θ.
In our simulations, we varied the initial size N from 100 to
40 000 and obtained 4320 to 14 400 realizations for eachcase (summarized in Table SI in the Supplemental Material[31]). For comparison, we also simulated flat interfaces, forwhich the initial condition was a line formed by 75 000
particles and the periodic boundary condition in the spanwise direction was used, and we obtained 14 400
realizations.
To characterize statistical properties of the stochastic
variable χðX; tÞ, we first estimated the nonuniversal parameters, v∞ , Γ, and A, from the data for the flat interfaces. v∞
and Γ were obtained by the standard procedure [4]—specifically, by using ∂ t hhi ≃ v∞ þ const × t−2=3 andhh2 ic =ðt2=3 hχ 1 2 ic Þ ≃ Γ2=3 , where h  k ic denotes the kthorder cumulant, and here we used the fact that theasymptotic fluctuations of the flat interfaces are given by
the GOE Tracy-Widom distribution. We obtained v∞ ¼
0.51370ð5Þ and Γ ¼ 0.980ð3Þ. The parameter A was
(a)

(b)

FIG. 1. Typical snapshots from the Eden simulations and theliquid-crystal experiments. (a) An Eden interface growing outward from a ring with N ¼ 1000 (dotted line). Time is indicatedby the color. (b) A DSM2 cluster (black) growing from a ringwith R0 ¼ 366 μm (dotted lines). The elapsed time after shooting
the laser is indicated above each image. The scale bar corresponds
to 1 mm.

060601-2

PHYSICAL REVIEW LETTERS 124, 060601 (2020)
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
obtained by A ¼ 2Γ=v∞, the relationship valid for
isotropic growth [11].
With those parameter values, we define the rescaledheightqðθ; tÞ ≔Rðθ; tÞ − v∞ t d≃ χð0; tÞðΓtÞ1=3

ð4Þ

and measure its mean and variance as functions of time, fordifferent initial particle numbers N (left panel of Fig. 2).
Figure 2 also shows the rescaled mean velocity [4,12]hpðθ; tÞi ≔



3t2=3
½∂ t Rðθ; tÞ − v∞ 
Γ1=3

≃ hχð0; tÞi þ 3t∂ t hχð0; tÞi;

ð5Þwhich asymptotically goes to hχð0; tÞi if hχð0; tÞi converges sufficiently fast. For the flat case (gray circles),hqi → hχ 1 i, hpi → hχ 1 i, and hq2 ic → hχ 1 2 ic as expected.
In the case of the ring initial conditions, for large N, thedata first behave similarly to the flat case, then deviate andapproach the values for the circular subclass, hχ 2 i and hχ 22 ic[37]. This crossover takes place earlier for smaller N.
Indeed, when the data are plotted against the rescaled timeτ ¼ v∞ t=R0 (R0 ¼ N=2π), all data collapse onto a single
curve except for the nonuniversal short-time regime (rightpanel of Fig. 2). This suggests that the distribution of χð0; tÞ
for different R0 ’s, denoted by χð0; t; R0 Þ, is described by asingle stochastic variable χ c ð0; τÞ, parametrized by τ, asfollows:dχð0; t; R0 Þ ⟶ χ c ð0; τÞ;ðR0 ; t → ∞Þ;

ð6Þ

where the double limit is taken with fixed τ ¼ v∞ t=R0 .
Then the flat-to-circular crossover we found indicatesddχ c ð0; τÞ ! χ 1 for τ → 0 and χ c ð0; τÞ ! χ 2 for τ → ∞.
The skewness Sk½Rðθ; tÞ ≔ hR3 ic =hR2 i3=2
→ Sk½χ c ð0; τÞcand the kurtosis Ku½Rðθ; tÞ ≔ hR4 ic =hR2 i2c → Ku½χ c ð0; τÞshow consistent behavior (see Fig. S1 in the Supplemental
Material [31]).
We also study this crossover in the spatial correlation.
In the case of the point initial condition, suppose
θ ¼ 0 corresponds to x ¼ 0. Then, using Rðθ; tÞ ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
hðx; tÞ2 þ x2 ≃ h þ x2 =2h and Eq. (2), we can showdqðθ; tÞ ! A2 ðXÞ. Therefore, the rescaled spatial covariance,Cs ðΔX; tÞ ≔ hqðθ þ Δθ; tÞqðθ; tÞi − hqðθ; tÞi2 with ΔX ≔hRðθ; tÞiΔθ=ξðtÞ, can be directly compared with the covariance of the Airy1 and Airy2 processes. Our numerical results
for the ring initial conditions (filled symbols in Fig. S2 of the
Supplemental Material [31]) indeed show crossover fromthe Airy1 covariance (τ ≪ 1) to the Airy2 covariance(τ ≫ 1), consistent with the results on the one-point
distribution.

FIG. 2. The mean and variance of the rescaled height, hqðθ; tÞiand hqðθ; tÞ2 ic , and the rescaled mean velocity, hpðθ; tÞi, for the
Eden model in the outgrowing case. The data are shown against
the raw time t (left) and the rescaled time τ ¼ v∞ t=R0 (right). The
theoretical curves evaluated numerically from the variational
formula for the outgrowing interfaces (¼ var. formula, blue solidline) are shown in the right panels. The values of χ 1 and χ 2 areshown by the dashed and dotted lines, respectively. The inset of
the top-right figure shows the difference between the data and theexcepted long-time limit value, hχ 2 i. The black solid line
indicates slope −1=3.

To test the universality of our finding, and in particular,
the function forms of hχ c ð0; τÞi and hχ c ð0; τÞ2 ic , weconducted experiments on liquid-crystal turbulence[4,10–12]. As in the previous studies, we applied an ac
voltage (here, 22 V at 300 Hz) to a nematic liquid crystal
filling a thin gap between transparent electrodes, and we
observed the growth of a turbulent state called dynamic
scattering mode 2 (DSM2), expanding in a metastableturbulent state, DSM1 (see the Supplemental Material [31]for detailed methods). DSM2 was generated by emitting a
few ultraviolet laser pulses [4]. Using the holographic
technique we previously adopted for the DSM2 growthexperiments [12], we formed the laser intensity profile inthe shape of a ring of a given radius R0 , which sets theinitial condition of the DSM2 interface [Fig. 1(b)]. We also
generated circular interfaces with a point initial condition,and flat interfaces with a linear initial condition. We
obtained 941 to 1936 realizations for each case (seeTable SII in the Supplemental Material [31]), recorded
by a charge-coupled device camera. The radius Rðθ; tÞ ofthe DSM2 interfaces [or the height hðx; tÞ for the flat case]was determined from each image, with the time t defined as
the elapsed time after shooting the laser pulses. Then the
nonuniversal parameters v∞ , Γ, and A were evaluated in thesame way as for the Eden model, here for the flat and point
initial conditions (Table SII [31]). Although the values of060601-3

PHYSICAL REVIEW LETTERS 124, 060601 (2020)
τ ¼ v∞ t=R0 . But why so, and how can this crossover betheoretically described? To answer these questions, weemploy the variational formula [16,23–26] and apply it to a
general, curved initial condition.
The variational formula describes the height hðx; tÞ for ageneral initial condition hðx; 0Þ ≕ h0 ðxÞ as follows:d

FIG. 3. Comparison of the results from the experiments (colorfilled symbols), the Eden simulations (gray open symbols), andthe variational formula (¼ var. formula, blue solid line), for theoutgrowing interfaces. The variance of the rescaled height,hqðθ; tÞ2 ic , and the rescaled mean velocity hpðθ; tÞi are shownin the left and right panels, respectively, against τ ¼ v∞ t=R0 . For
the numerical results, data with t > 103 are shown by the same
symbols as those in Fig. 2. For the experimental results, statisticalerrors are indicated by the error bars on the first and last data
points, and uncertainty associated with the parameter estimationis shown by the shaded areas. The values for χ 1 (flat) and χ 2(circular) are shown by the dashed and dotted lines, respectively.
The inset of the right panel shows the experimental results
obtained with v∞ from the flat case, while it was adjusted in themain panel to fit the Eden data at the largest t (see text).

v∞ , Γ, and A are expected to be independent of the initialcondition, in practice one needs to evaluate for each set ofexperiments, because of unavoidable slight changes inexperimental conditions [11]. For the ring initial conditions, however, the parameter values could not be obtainedin the same way because of the time dependence (i.e.,crossover) of χðX; tÞ. We therefore used the values obtained
from the flat case for the outgrowing cases, unless otherwise stipulated. Possible shifts in the parameter values weretaken into account in the uncertainty estimates for theoutgrowing cases, evaluated from the differences in theparameter values between the flat and circular cases.
Now, we compare the experimental results with those for
the Eden model. The left panel of Fig. 3 shows the varianceof the rescaled height, hqðθ; tÞ2 ic , against τ ¼ v∞ t=R0 ,which overlaps on the Eden data within statistical errors
and parameter uncertainty (error bars and shades, respectively) apart from the nonuniversal short-time behavior. For
the rescaled mean velocity hpðθ; tÞi (right panel), theuncertainty of v∞ was too large to make a meaningfulcomparison (inset). However, if we instead choose thevalue of v∞ in such a way that hpðθ; tÞi at the largest t falls
onto the curve for the Eden model (obtained values of v∞
are given in Table SII [31]), hpðθ; tÞi overlaps for all t(main panel). Those results of hqðθ; tÞ2 ic and hpðθ; tÞisuggest universality of the one-point distribution ofχ c ð0; τÞ. Moreover, the spatial covariance Cs ðΔX; tÞ is also
found to overlap with the results of the Eden model if thevalue of τ is close enough (Fig. S2 [31]). This suggests that
not only the one-point distribution of χ c ð0; τÞ, but also thespatial covariance of χ c ðX; τÞ, is universal.
So far, we have characterized the flat-to-circular crossover and found it to be controlled by a single parameter

hðx; tÞ ≃ sup ½hcirc ðx; t; yÞ þ h0 ðyÞ;

ð7Þ

y∈R

where hcirc ðx; t; yÞ denotes the height for the point initialcondition nucleating at position y, growing with the same
realization of noise for different y [23]. Intuitively, thismeans that the initial condition hðx; 0Þ can be regarded as acollection of point sources, and hðx; tÞ is then given by theenvelope of the circular interfaces from those point sources,a bit analogously to Huygens’ principle [38]. The formula (7) involves a mathematical object called the Airy
sheet [23,25], but if the interest is only in the one-point
distribution, it can be simply expressed by the Airy2
process, as follows [16,24]:h0 ðξðtÞYÞ
d2χðX; tÞ ≃ sup A2 ðX − YÞ − ðX − YÞ þ
: ð8ÞðΓtÞ1=3
Y∈R
We use Eq. (8) and consider a class of curved initialconditions in the following form:xh0 ðxÞ ¼ R0 g;
ð9Þ
R0where gðwÞ is a locally parabolic function, i.e., gðwÞ ¼−c2 w2 þ Oðw2 Þ for small jwj. Substituting Eq. (9) into
Eq. (8), taking the limit R0 , t → ∞ with fixed τ ¼ v∞ t=R0 ,and setting x ¼ 0 yieldsdχð0; tÞ ⟶ sup½A2 ðYÞ − ð1 þ cτÞY 2  ≕ χ̃ðcτÞ;

ð10Þ

Y∈R

with c ≔ ð4c2 ΓÞ=ðA2 v∞ Þ. This shows that the asymptotic
height distribution is parametrized only by cτ, and only the
local functional form of gðwÞ at small jwj is relevant.
The characteristic time is τ ¼ 1=c in the rescaled unit,
and therefore t ¼ A2 R0 =4c2 Γ in the raw time scale, andthis is the time at which the initial height difference
jh0 ð0Þ − h0 ðξðtÞÞj becomes comparable to the fluctuation
1=3
amplitude,ðΓtÞ. For isotropic growth, the relationshipﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A ¼ 2Γ=v∞ [11] further yields c ¼ 2c2 .
For the ring initial conditions, gðwÞ is given by gðwÞ ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
σð 1 − w2 1jwj<1 − 1Þ with σ ¼ þ1 (−1) for the outgrowding (ingrowing) case. Then we obtain χð0; tÞ ≃ χ̃ðστÞ,
which we have expressed by χ c ð0; τÞ for the outgrowingcase σ ¼ þ1 [Eq. (6)]. Note that, mathematically, it isdknown that χ̃ð0Þ ¼ supY∈R ðA2 ðYÞ − Y 2 Þ ¼ χ 1 ; i.e., theGOE Tracy-Widom distribution [39,40]. In the other limit

060601-4

PHYSICAL REVIEW LETTERS 124, 060601 (2020)dτ → ∞, clearly, χ̃ðτÞ → A2 ð0Þ ¼ χ 2 ; i.e., the GUE TracyWidom distribution. Therefore, χ c ð0; τÞ ¼ χ̃ðτÞ indeed has
the expected limits on both sides of the flat-to-circularcrossover.
To compare the variational formula with the experimental and numerical data for finite τ, we employ aMonte Carlo method to evaluate Eq. (10). The Airy2
process A2 ðYÞ is in fact known to be equivalent to thelargest eigenvalue of large GUE random matrices undergoing Dyson’s Brownian motion [16,39]. We therefore
implement Dyson’s Brownian motion numerically, in theform of the Ornstein-Uhlenbeck process of Hermitian
random matrices and obtain approximated realizations of
A2 ðYÞ (see the Supplemental Material [31] for details).
Then, we evaluate the supremum of Eq. (10), interpolating
the values of A2 ðYÞ between the discrete steps by using theBrownian bridge [31]. The results for the outgrowing case(σ ¼ þ1) are shown in Figs. 2 and 3, where the data of themean hqi, the variance hq2 ic , and the rescaled mean
velocity hpi are compared with the corresponding expressions of χ̃ðτÞ—specifically, hχ̃ðτÞi, hχ̃ðτÞ2 ic [Eq. (4)], andhχ̃ðτÞi þ 3τ∂ τ hχ̃ðτÞi [Eq. (5)], respectively. The results ofthe variational formula precisely agree, without any adjustable parameter, with the numerical and experimental data.
We also inspected the ingrowing case σ ¼ −1 and confirmed the validity of the variational formula (see Fig. S3 of
the Supplemental Material [31]). The agreement was also
underpinned for the skewness and kurtosis (Fig. S4 of the
Supplemental Material [31]).
In summary, we found KPZ crossover functions thatgovern height fluctuations of interfaces growing outward
from ring initial conditions, parametrized only by therescaled time τ ¼ v∞ t=R0 , and evidenced their universality
both experimentally and numerically. We then presented atheoretical description of this crossover, on the basis of the
KPZ variational formula for general curved initial conditions. We numerically evaluated the formula and foundremarkable agreement with the experimental and numerical
data. Our results constitute the first example where the KPZvariational formula was successfully used to describe
experimental observations, showing the ability of thisformula to explain, or even predict, real data from general
initial conditions. We hope our work will trigger further
studies to elucidate geometry-dependent universality of the
KPZ class and beyond.
We thank P. Le Doussal for useful discussions on thevariational formula, and F. Bornemann for the theoreticalcurves of the Airy1 and Airy2 covariance [41]. We thank theSupercomputer Center of the Institute for Solid State
Physics (the University of Tokyo) and the Meiji Institutefor Advanced Study of Mathematical Sciences (Meiji
University) for computational facilities. We acknowledge
financial support by KAKENHI from the Japan Society for
the Promotion of Science (Grants No. JP25103004,

No. JP16H04033, No. JP19H05800, No. JP19H05144,
No. JP17J05559), by the Yamada Science Foundation,and by the National Science Foundation (Grant No. NSF
PHY11-25915).

*

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