PHYSICAL REVIEW LETTERS 124, 250602 (2020)

Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces
1

Takayasu Iwatsuka,1,2 Yohsuke T. Fukai ,3,2 and Kazumasa A. Takeuchi

2,1,*Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan2
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
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, Japan(Received 24 April 2020; accepted 29 May 2020; published 24 June 2020)
The nonequilibrium steady state of the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) universality
class has been studied in-depth by exact solutions, yet no direct experimental evidence of its characteristic
statistical properties has been reported so far. This is arguably because, for an infinitely large system,
infinitely long time is needed to reach such a stationary state and also to converge to the predicted universal
behavior. Here we circumvent this problem in the experimental system of growing liquid-crystal
turbulence, by generating an initial condition that possesses a long-range property expected for the
KPZ stationary state. The resulting interface fluctuations clearly show characteristic properties of the 1D
stationary KPZ interfaces, including the convergence to the Baik-Rains distribution. We also identify finitetime corrections to the KPZ scaling laws, which turn out to play a major role in the direct test of thestationary KPZ interfaces. This paves the way to explore unsolved properties of the stationary KPZinterfaces experimentally, making possible connections to nonlinear fluctuating hydrodynamics andquantum spin chains as recent studies unveiled relation to the stationary KPZ.
DOI: 10.1103/PhysRevLett.124.250602

Introduction.—The Kardar-Parisi-Zhang (KPZ) universality class describes dynamic scaling laws of a variety ofphenomena, ranging from growing interfaces to directed
polymers and stirred fluids [1,2], as well as fluctuatinghydrodynamics [3] and, most recently, quantum integrable
spin chains [4], to name but a few. The KPZ class is now
central in the studies of nonequilibrium scaling laws,
mostly because some models in the one-dimensional(1D) KPZ class turned out to be integrable and exactly
solvable (for reviews, see, e.g., [5,6]). This has unveiled awealth of nontrivial fluctuation properties in such nonequilibrium and nonlinear many-body problems.
The KPZ class is often characterized by the KPZ
equation, a paradigmatic model for interfaces growing influctuating environments [1,2,5]. It reads, in the case of 1Dinterfaces in a plane: ∂∂ 2 h λ ∂h 2
hðx; tÞ ¼ ν 2 þþ ηðx; tÞ:
∂t
2 ∂x
∂x

ð1Þ

Here hðx; tÞ denotes the position of the interface in the
direction normal to a reference line (e.g., substrate), often
called the local height, at lateral position x and time t.
ηðx; tÞ is white Gaussian noise with hηðx; tÞi ¼ 0 andhηðx; tÞηðx0 ; t0 Þi ¼ Dδðx − x0 Þδðt − t0 Þ, where h  i denotes
the ensemble average. Such random growth develops
nontrivial fluctuations of hðx; tÞ, characterized by a setof universal power laws. For example, the fluctuation
0031-9007=20=124(25)=250602(6)amplitude of hðx; tÞ grows as tβ , with β ¼ 1=3 for 1D.
This implies
hðx; tÞ ≃ v∞ t þ ðΓtÞ1=3 χ þ Oðt0 Þ;

ð2Þwith constant parameters v∞ ; Γ and a rescaled randomvariable χ. χ is correlated in space and time but characterized by a distribution that remains well defined in thelimit t → ∞. Another important quantity is the heightdifference correlation function, defined by Ch ðl; tÞ≡h½hðx þ l; tÞ − hðx; tÞ2 i. While Ch ðl; tÞ ∼ t2β for l muchlarger than the correlation length ξðtÞ ∼ t1=z , for l ≪ ξðtÞ,Ch ðl; tÞ ∼ l2α with α ¼ zβ [2,5]. For 1D, the scaling
exponents are α ¼ 1=2; β ¼ 1=3; z ¼ 3=2 and shared
among members of the KPZ universality class [1,2,5,6].
Moreover, for the 1D KPZ equation (1), the (statistically)
stationary state of this particular model, hKPZeq
ðxÞ, isstatknown to be equivalent to the 1D Brownian motion[1,2,5,6]:
hKPZeq
ðxÞ ¼
stat

pﬃﬃﬃﬃ
ABðxÞ:

ð3Þ

Here, A ≡ D=2ν and BðxÞ is the standard Brownian motion
with time x, so that hBðxÞi ¼ 0 and h½Bðx þ lÞ−BðxÞ2 i ¼ l. The height-difference correlation function for
hKPZeqðxÞ is then simply the mean-squared displacement,
stat
ChKPZeq ðlÞ ≃ Al, with A corresponding to the diffusion
stat

ðxÞ,
coefficient. Note that, even if we set hðx; 0Þ ¼ hKPZeq
stat

250602-1© 2020 American Physical Society

PHYSICAL REVIEW LETTERS 124, 250602 (2020)
hðx; tÞ still fluctuates and grows, i.e., hhðx; tÞi ¼ v∞ t witha constant v∞ . Nevertheless, the shifted height hðx; tÞ −v∞ t can be always described by Eq. (3) with anotherinstance of BðxÞ (which is actually correlated with the oneused for the initial condition). For lack of a better term, here
we call it the (statistically) stationary state of the KPZ
equation.
Then the exact solutions of the 1D KPZ equation [7–13],
as well as earlier results for discrete models (e.g., [14,15]),
unveiled detailed fluctuation properties of hðx; tÞ, inparticular the distribution of χ [5,6]. Further, those properties turned out to depend on the global geometry ofinterfaces or on the initial condition hðx; 0Þ, being classified into a few universality subclasses within the single
KPZ class. Among them, most important and establishedare the subclasses for circular, flat, and stationary interfaces, characterized by the following asymptotic distributions [5]: the GUE Tracy-Widom [16], GOE Tracy-Widom[17], and Baik-Rains distributions [18], respectively (GUEand GOE stand for the Gaussian unitary and orthogonalensembles, respectively). More precisely, with the random
numbers drawn from those distributions, denoted by χ 2, χ 1 ,d

χ 0 [19], respectively, we have χ!χ 2 ; χ 1 ; χ 0 for the threedrespective subclasses [20], where ! indicates the convergence in the distribution. For the KPZ equation, thetypical initial conditions that correspond to the threesubclasses are hðx; 0Þ ¼ −jxj=δ ðδ → 0þ Þ (circular),pﬃﬃﬃﬃ
hðx; 0Þ ¼ 0 (flat), and hðx; 0Þ ¼ hKPZeq
ðxÞ ¼ ABðxÞ (stastat
tionary). Experimentally, the circular and flat subclasses
were clearly observed in the growth of liquid-crystal
turbulence [5,21,22], but only indirect and partial support
has been reported so far for the stationary subclass [23,24](see also [25]). This is presumably because, firstly, for an
infinitely large system, it takes infinitely long time for asystem to reach the stationary state [as ξðtÞ ∼ t2=3 needs toreach infinity]. Then one should take an interface profile inthe stationary state, regard it as an “initial condition,” andwait sufficiently long time for the height fluctuations toconverge to the Baik-Rains distribution (see Ref. [23] formore quantitative arguments). For a finite system of size L,
reaching the stationary state takes a finite time ∼L3=2 , but
the approach to the Baik-Rains distribution is now visible
only within a finite time period [27,28], being eventuallyreplaced by a final state unrelated to the choice of the initialcondition.
Here we overcome this difficulty in the liquid-crystal
experimental system, by generating an interface that
resembles the expected stationary state. Using a holographic technique developed previously [29], we generated
Brownian initial conditions (3) for the growing turbulenceand directly measured fluctuation properties of the height
hðx; tÞ under this type of initial conditions [Fig. 1(b)]. This
allowed us to carry out quantitative tests of a wealth ofexact results for integrable models in the stationary state.

(a)

(b)

FIG. 1. Typical snapshots of a flat (a) and a Brownian (b) interface, separating the metastable DSM1 (gray) and growing DSM2
regions (black). hlab ðx; tlab Þ denotes the position of the upperinterface in the laboratory frame, at time tlab from the laser
emission. t and hðx; tÞ are defined as follows: t ≡ tlab andhðx; tÞ ≡ hlab ðx; tlab Þ − hhðx; tinitlab Þix for the flat case (a), t ≡ tlab −
init
tinit
lab and hðx; tÞ ≡ hðx; tlab Þ − hðx; tlab Þ for the Brownian case (b).
See also Movies S1 and S2 [30].

And indeed, we obtained direct evidence for the Baik-Rains
distribution and the related correlation function. This opens
an experimental pathway to explore universal yet hitherto
unsolved statistical properties of the KPZ stationary state.
Methods.—The experimental system was a minor modification of that used in Ref. [29] (see Section I of the
Supplemental Material and Fig. S1 [30] for details). We useda standard material for the electroconvection of nematic
liquid crystal [32], specifically, N-(4-methoxybenzylidene)4-butylaniline doped with tetra-n-butylammonium bromide.
The liquid crystal sample was placed between two parallel
glass plates with transparent electrodes, separated by spacersof thickness 12 μm. The electrodes were surface-treated torealize homeotropic alignment. The temperature was maintained at 25 °C during the experiments, with typical fluctuations of 0.01 °C.
The electroconvection was induced by applying an ac
voltage to the system. In this Letter, we fixed the frequency
at 250 Hz, well below the cutoff frequency near 1.8 kHz,and the voltage was set to be 23 V. At this voltage, thesystem is initially in a turbulent state called the dynamic
scattering mode 1 (DSM1), which is actually metastable, sothat the stable turbulent state DSM2 eventually nucleatesand expands, forming a growing cluster bordered by afluctuating interface. One can also trigger DSM2 nucleation by shooting an ultraviolet (UV) laser pulse [5]. Thisnot only allows us to carry out controlled experiments but
also to design the initial shape of the interface, by changing
the intensity profile of the laser beam. Growing interfaces
were observed by recording light transmitted through the250602-2

PHYSICAL REVIEW LETTERS 124, 250602 (2020)dhhi
Γ1=3 hχi −2=3:
≃ v∞ þ
tdt
3

ð4Þ

Therefore, reading the y intercept of linear regression,
we obtained v∞ ¼ 36.86ð4Þ μm=s, where the numbersin the parentheses indicate the uncertainty. For Γ, since
the flat interfaces in this liquid-crystal system were alreadyshown to exhibit the GOE Tracy-Widom distribution[5,21,22], we have hhn ic ≃ ðΓtÞn=3 hχ n1 ic ðn ≥ 2Þ, where
hXn ic denotes the nth-order cumulant of a variable X.

36
35
34

( m2 s-2/3)
86
83
80

33
0

h2 t -2/3
c

t
0

0.3

t

-2/3

-2/3

0.6 (s)0.5-2/3 -2/3

(s

)

1

qðx; tÞ ≡

(a)

4s
15s
50s

10

BR

6

10

-5

0

q

Brownian 10

100

(b)

flat

4s
15s
50s
0 0
10
102
5

8

GOE TW-5

hðx; tÞ − v∞ t≃ χ:ðΓtÞ1=3

ð5Þ

Clear agreement with the GOE Tracy-Widom distribution
was confirmed [Fig.p2(b)].
those estimates, we
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃUsing
ﬃ
finally obtained A ¼ 2Γ=v∞ ¼ 8.762ð13Þ μm.
Brownian interface experiments.—Based on the value ofA evaluated by the flat interface experiments, we generated
Brownian initial conditions (3) with A ¼ 9 μm [36] andstudied growing DSM2 interfaces [Fig. 1(b)]. Each initialcondition was prepared by projecting a hologram of acomputer-generated Brownian trajectory, with resolution of
36.5 μm at the liquid-crystal cell, by using a spatial light
modulator [30]. The height profile in the laboratory frame
hlab ðx; tlab Þ was determined as for the flat experiments, buthere the height of interest is the increment from the heightprofile at the first analyzable time, hðx; tÞ ≡ hlab ðx; tlab Þ−
init
init
hlab ðx; tinitlab Þ, with t ≡ tlab − tlab and tlab ¼ 0.2 s [Fig. 1(b)].
We used a region of width 2730 μm near the center of thecamera view and analyzed 1021 interfaces. Finite-size
effect is expected to be prevented, because the Brownian
trajectories were much longer (4670 μm in x) than thewidth of the analyzed region.
First we test whether the interfaces generated thereby arestationary or not. To this end, we measure the heightdifference correlation function for hlab ðx; tlab Þ, Chlab ðl; tlab Þ,
and find that it does depend on tlab [Fig. 3(a)], indicatingthat the interfaces are not stationary. More precisely, we12(b) 100
prob. density

d h /dt ( m/s)(a) 37

Above all, the variance can be most precisely determined,
and is known to grow, with the leading finite-time correction, as hh2 ic ≃ ðΓtÞ2=3 hχ 21 ic þ Oðt0 Þ [5,22,34]. Therefore,by plotting hh2 ic t−2=3 against t−2=3 [Fig. 2(a), inset] andreading the y intercept of linear regression, we obtained
Γ ¼ 1415ð4Þ μm=s [35]. Consistency was checked byplotting the histogram of the height, rescaled with thoseparameters as follows

4

10

0.2s
1s
4s
50s

102

104

d h /dt ( m/s)

sample, using a light-emitting diode as the light source anda charge-coupled device camera.
Flat interface experiments.—In order to realize
Brownian initial conditions (3) that may correspond to
the stationary state, we first need to evaluate the parameter
A. To this end we first carried out a set of experiments forflat interfaces. Using a cylindrical lens to expand the laser
beam, we generated an initially straight interface for eachexperiment and tracked growth of the upper interface[Fig. 1(a)]. The h axis is set along the mean growth
direction. The x axis is normal to h, along the initialstraight line. Then the coordinates of the upper interface inthe laboratory frame were extracted and denoted byhlab ðx; tlab Þ, where tlab is the time elapsed since the laser
emission. Since the height of interest is the increment fromthe initial interface, we approximated it by the spatiallyaveraged height at the first analyzable time, denoted byinit
hhðx; tinitlab Þix , with tlab ¼ 0.2 s. Then we defined hðx; tÞ ≡hlab ðx; tlab Þ − hhðx; tinitlab Þix with t ≡ tlab and studied its
fluctuations over 1267 independent realizations. In thefollowing, the ensemble average h  i was evaluated byaveraging over all realizations and spatial points x.
Theparameter
A can be determined by the relation
ﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A ¼ 2Γ=v∞ , known to hold in isotropic systems [5,22].
For v∞, we followed the standard procedure [5,33] andplotted dhhi=dt against t−2=3 [Fig. 2(a), main panel]. From
Eq. (2), we haved h /d t ( m/s)
37

37

36

35
0

36
35
0

0.5

t-2/3

0.5

t-4/3 (s-4/3)

1 (s-2/3)

1

5

FIG. 2. Parameter estimation for the flat interfaces. (a) dhhi=dt
against t−2=3 (main panel) and hh2 ic t−2=3 against t−2=3 (inset). The
dashed lines show the results of linear regression. (b) Histogramsof the rescaled height qðx; tÞ at different t (legend). Agreementwith the GOE Tracy-Widom (TW) distribution is confirmed. BR
stands for the Baik-Rains distribution.

FIG.
3. EvaluationoftheBrownianinterfaces.
(a) Chlab ðl; tlab Þ=l against l for different tlab (indicated in thelegends) for the Brownian (main panel) and flat (inset) interfaces.
The black stars indicate the results of direct evaluation of
the computer-generated images used for the holograms.
(b) dhhi=dt against t−4=3 (main panel) and t−2=3 (inset) for the
Brownian interfaces. The dashed line in the main panel shows theresult of linear regression.

250602-3

PHYSICAL REVIEW LETTERS 124, 250602 (2020)
100

4s
15s
49.8s
BR

(b) 3

-2

0

q

2

1
0

GOE TW

10-5 -4

4s
15s
49.8s
exact

2

g''(y)

prob. density

(a)

4

-1-4

6

(c)

-2

1

4s
15s
49.8s
exact

0.5

0

0

0

2

4

P (t1,t 2) t 2/3(s 2/3)

0.3

(d)

0.2
0

10

P (t1, t2)

Ct(t1, t2) / Ct(t2, t2)

observe that Chlab ðl; tlab Þ=l at small l initially takes values
lower than the desired one, A ¼ 9 μm, presumably becauseof the finite resolution of the holograms, then increases up
to ≈11 μm. The fact that Chlab ðl; tlab Þ=l becomes higher
than A at small l was also observed in our flat data
[Fig. 3(a) inset] as well as in our past experiments [5,21].
However, more important is the behavior at large l, whichturns out to be stable and takes a value close to A ¼ 9 μm.
Therefore, in the following we test whether our interfaces,though not stationary, can nevertheless exhibit universal
properties of the stationary KPZ subclass, such as the BaikRains distribution.
To determine the scaling coefficients, we plot dhhi=dt
against t−2=3 in the inset of Fig. 3(b). Time dependence of
dhhi=dt confirms nonstationarity of the interfaces again.
Interestingly, as opposed to the result for the flat interfaces[Fig. 2(a)], here we do not find linear relationship to t−2=3[Fig. 3(b), inset], but to t−4=3 (main panel). From Eq. (4),this suggests hχi ¼ 0, consistent with the vanishing meanof the Baik-Rains distribution hχ 0 i ¼ 0. If so, the subleading term of Eq. (4) is indeed expected to be Oðt−4=3 Þ,coming from a t−1=3 term expected to exist in Eq. (2). Then,by linear regression, we obtained v∞ ¼ 37.126ð15Þ μm=s.
It is reasonably close to the value from the flat experiments,
in view of the typical magnitude of parameter shifts in thisexperimental system [22]. For Γ, we took the value from theflat experiments, so that we do not make any assumption on
the statistical properties for the Brownian case.
Using the values of v∞ and Γ determined thereby, as wellpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
as A ¼ 2Γ=v∞ , we test various predictions for thestationary KPZ subclass, without any adjustable parameter.
The results are summarized in Fig. 4. Figure 4(a) shows
histograms of the rescaled height qðx; tÞ [Eq. (5)] at
different times t. The obtained distributions at finite times
are already close to the predicted Baik-Rains distribution.
Indeed, convergence in the t → ∞ limit is confirmed
quantitatively by analyzing finite-time corrections in the
cumulants (Section II of the Supplemental Material andFig. S2 [30]). In Fig. 4(b), we test the prediction on the twopoint correlation function C2 ðl; tÞ ≡ h½hlab ðl þ x; t þ t0 Þ−
hlab ðx; t0 Þ − v∞ t2 i. It is often denoted by gðyÞ in therescaled units, with y ≡ l=ξðtÞ, ξðtÞ ≡ ð2=AÞðΓtÞ2=3 , andgðyÞ ≡ ðΓtÞ−2=3 C2 ðl; tÞ. Its second derivative, g00 ðyÞ, plays
the pivotal role in the emergence of KPZ in fluctuatinghydrodynamics [3] and quantum integrable spin chains [4].
This is tested with our experimental data and good agreement is found [Fig. 4(b)]. Figure 4(c) shows the resultsof the two-time correlation of hðx; tÞ, Ct ðt1 ; t2 Þ ≡hδhðx; t1 Þδhðx; t2 Þi with δhðx; tÞ ≡ hðx; tÞ − hhðx; tÞi.
Our data agree with Ferrari and Spohn’s prediction [38]that the two-time correlation coincides with that of thefractional Brownian motion with Hurst exponent 1=30.5

t 1 / t2

1

0.1 -1
10
P+ P ( 5)

10-2 -1
10

4s
15s
30s

t (s)
0

10

1

10

t -2/3100

101

t = t2t1 (s)

102

FIG. 4. Main results of the Brownian interface experiments.
(a) Histograms of the rescaled height qðx; tÞ at different t(legend). The data are found to converge to the Baik-Rains(BR) distribution, as shown quantitatively in Section II of the
Supplemental Material and Fig. S2 [30]. GOE TW stands for the
GOE Tracy-Widom distribution. (b) Two-point correlation function g00 ðyÞ. The experimental data are evaluated by ½ξðtÞ2 =initðΓtÞ2=3 2hð∂hlab =∂xÞðl þ x; t þ tinitwithlab Þð∂hlab =∂xÞðx; tlab Þidifferent t (legend). The black curve indicates Prähofer andSpohn’s exact solution [40,41]. (c) Rescaled two-time function
Ct ðt1 ; t2 Þ=Ct ðt2 ; t2 Þ for different t2 (legend). The data are foundto converge to Ferrari and Spohn’s exact solution [38] (black
curve), as shown quantitatively in Section III of the Supplemental
Material and Fig. S3 [30]. (d) Persistence probability P ðt1 ; t2 Þ
for different t1 (legend). For visibility, P− ðt1 ; t2 Þ is shifted byfactor 5. The dashed line is a guide for eyes indicating the powerlaw t−2=3 for FBM1=3. The inset shows P ðt1 ; t2 ÞΔt2=3 .

(hereafter abbreviated to FBM1=3 ), Ct ðt1 ; t2 Þ=Ct ðt2 ; t2 Þ →ð1=2Þ½1 þ ðt1 =t2 Þ2=3 − ð1 − t1 =t2 Þ2=3  (black line) in thelimit t1 ; t2 → ∞ with fixed t1 =t2 (see Section III of the
Supplemental Material and Fig. S3 [30] for a quantitativetest). Finally, Fig. 4(d) shows the persistence probabilityP ðt1 ; t2 Þ; i.e., this is the probability that hðx; tÞ − hðx; t1 Þremains always positive (Pþ ) or negative (P− ) until time t2 ,which is found to decay clearly as P ðt1 ; t2 Þ ∼ Δt−2=3 withΔt ≡ t2 − t1 . The persistence exponent is therefore 2=3,
supporting Krug et al.’s conjecture [39] that it also
coincides with that of FBM1=3 . Those relations to
FBM1=3 are intriguing, because hðx; tÞ is not Gaussian
and therefore its time evolution is not FBM1=3 .
Concluding remarks.—In this Letter, we aimed atunambiguous tests of universal statistics for the stationary
state of the (1 þ 1)-dimensional KPZ class. Instead of
waiting for the interfaces to approach the stationary state,
we generated such initial conditions that are expected toshare the same long-range properties with the stationary
state, specifically, the Brownian initial conditions (3)with the appropriate diffusion coefficient A determined

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PHYSICAL REVIEW LETTERS 124, 250602 (2020)
beforehand. The resulting interfaces turned out to be notstationary, but nevertheless our data clearly showed thedefining properties of the stationary KPZ subclass, including the Baik-Rains distribution and the two-point correlation function g00 ðyÞ [Figs. 4(a) and 4(b)]. Our results also
support intriguing relations to time correlation properties ofthe fractional Brownian motion [Figs. 4(c) and 4(d)], which
may deserve further investigations in other quantities. With
this and past studies [5,21,22], all the three representative
KPZ subclasses in one dimension [5,6] were given experimental supports for the universality.
The KPZ class has been extensively studied already for
decades, yet it continues finding novel connections tovarious areas of physics (recall recent developments innonlinear fluctuating hydrodynamics [3] and quantum spin
chains [4]). We hope our experiments will also serve toprobe quantities of interest for those systems, which may be
not always solved exactly but still have a possibility to bemeasured precisely. Explorations of higher dimensions, forwhich numerics have played leading roles [6,42], are also
important directions left for future studies.
We thank P. L. Ferrari, T. Halpin-Healy, T. Sasamoto,and H. Spohn for enlightening discussions. We are also
grateful to M. Prähofer and H. Spohn for the theoreticalcurves of the BR and GOE-TW distributions and that of thestationary correlation function gðζÞ, which are madeavailable online [41]. This work is supported in part by
KAKENHI from Japan Society for the Promotion of
Science (Grants No. JP25103004, No. JP16H04033,
No. JP19H05144, No. JP19H05800, No. JP20H01826,
and No. JP17J05559), by Tokyo Tech Challenging
Research Award, by Yamada Science Foundation, andby the National Science Foundation (Grant No. NSF
PHY11-25915).

*

kat@kaztake.org[1] M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56,
889 (1986).
[2] A.-L. Barabási and H. E. Stanley, Fractal Concepts in
Surface Growth (Cambridge University Press, Cambridge,
England, 1995).
[3] H. Spohn, in Thermal Transport in Low Dimensions,Lecture Notes in Physics Vol. 921, edited by S. Lepri(Springer, Heidelberg, 2016), Chap. 3, pp. 107–158.
[4] M. Ljubotina, M. Žnidarič, and T. Prosen, Phys. Rev. Lett.
122, 210602 (2019); S. Gopalakrishnan and R. Vasseur,
Phys. Rev. Lett. 122, 127202 (2019); A. Das, M. Kulkarni,
H. Spohn, and A. Dhar, Phys. Rev. E 100, 042116(2019).
[5] K. A. Takeuchi, Physica (Amsterdam) 504A, 77 (2018).
[6] H. Spohn, in Stochastic Processes and Random Matrices,
Lecture Notes of the Les Houches Summer School Vol. 104,
edited by G. Schehr, A. Altland, Y. V. Fyodorov, N.
O’Connell, and L. F. Cugliandolo (Oxford University Press,
Oxford, 2017), Chap. 4, pp. 177–227; T. Halpin-Healy and

K. A. Takeuchi, J. Stat. Phys. 160, 794 (2015); I. Corwin,
Random Matrices Theory Appl. 01, 1130001 (2012).
[7] T. Sasamoto and H. Spohn, Phys. Rev. Lett. 104, 230602(2010).
[8] G. Amir, I. Corwin, and J. Quastel, Commun. Pure Appl.
Math. 64, 466 (2011).
[9] P. Calabrese, P. Le Doussal, and A. Rosso, Europhys. Lett.
90, 20002 (2010).
[10] V. Dotsenko, Europhys. Lett. 90, 20003 (2010).
[11] P. Calabrese and P. Le Doussal, Phys. Rev. Lett. 106,
250603 (2011).
[12] T. Imamura and T. Sasamoto, Phys. Rev. Lett. 108, 190603(2012).
[13] A. Borodin, I. Corwin, P. Ferrari, and B. Vető, Math. Phys.
Anal. Geom. 18, 20 (2015).
[14] K. Johansson, Commun. Math. Phys. 209, 437 (2000).
[15] M. Prähofer and H. Spohn, Phys. Rev. Lett. 84, 4882
(2000).
[16] C. A. Tracy and H. Widom, Commun. Math. Phys. 159, 151(1994).
[17] C. A. Tracy and H. Widom, Commun. Math. Phys. 177, 727(1996).
[18] J. Baik and E. M. Rains, J. Stat. Phys. 100, 523 (2000).
[19] With the standard GOE Tracy-Widom random variableχ 1;TW (as defined in Ref. [17]), χ 1 is defined by χ 1 ≡2−2=3 χ 1;TW [5].
[20] In the circular case, for x ≠ 0, an additional shift proportional to x2 =t is needed for the convergence to χ 2 , tocompensate the locally parabolic mean profile of theinterfaces [5,6]. In the stationary case, the left-hand sideof Eq. (2) should be more precisely hðx; tÞ − hðx; 0Þ, but by
imposing hð0; 0Þ ¼ 0 one can still use Eq. (2) at x ¼ 0 todshow χ!χ 0 [5,6].
[21] K. A. Takeuchi and M. Sano, Phys. Rev. Lett. 104, 230601(2010); K. A. Takeuchi, M. Sano, T. Sasamoto, and H.
Spohn, Sci. Rep. 1, 34 (2011).
[22] K. A. Takeuchi and M. Sano, J. Stat. Phys. 147, 853 (2012).
[23] K. A. Takeuchi, Phys. Rev. Lett. 110, 210604 (2013).
[24] K. A. Takeuchi, J. Phys. A 50, 264006 (2017).
[25] There was a claim for an observation of the Baik-Rains
distribution in an experiment of paper combustion [26], but
it seems to us that their precision is not sufficient todistinguish it from other possible distributions, as detailed
in the commentary article available at http://publ.kaztake.org/miet-com.pdf. Note also that in the stationary state of afinite-size system, as studied in this experiment, an approachto the Baik-Rains distribution will appear in a finite time
window, so that careful analysis of time dependence is
crucial.
[26] L. Miettinen, M. Myllys, J. Merikoski, and J. Timonen,
Eur. Phys. J. B 46, 55 (2005).
[27] S. Prolhac, Phys. Rev. Lett. 116, 090601 (2016).
[28] J. Baik and Z. Liu, Commun. Pure Appl. Math. 71, 747(2018).
[29] Y. T. Fukai and K. A. Takeuchi, Phys. Rev. Lett. 119,030602 (2017); 124, 060601 (2020).
[30] See the Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.124.250602 for additional text (Sections I–III), Figs. S1–S3, and Movies S1and S2. Ref. [31] is cited therein.

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PHYSICAL REVIEW LETTERS 124, 250602 (2020)[31] J. De Nardis, P. Le Doussal, and K. A. Takeuchi, Phys. Rev.
Lett. 118, 125701 (2017).
[32] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,
2nd ed., International Series of Monographs on Physics
Vol. 83 (Oxford Univ. Press, New York, 1995).
[33] J. Krug, P. Meakin, and T. Halpin-Healy, Phys. Rev. A 45,
638 (1992).
[34] P. L. Ferrari and R. Frings, J. Stat. Phys. 144, 1123 (2011).
[35] Although Γ can also be estimated from Eq. (4), reading theslope of Fig. 2(a) is much less precise than the estimation
based on the variance.
[36] To reduce the effect of parameter shift, we chose to start the
Brownian interface experiments before completing thecareful analysis of the flat experimental data. As a result,
we used a rough estimate A ¼ 9 μm for the Brownian initialconditions. A slight difference in A is expected to have only
a minor impact on the fluctuation properties of hðx; tÞ [37].
[37] S. Chhita, P. L. Ferrari, and H. Spohn, Ann. Appl. Probab.
28, 1573 (2018).
[38] P. L. Ferrari and H. Spohn, SIGMA 12, 074 (2016).
[39] J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J.
Bray, and C. Sire, Phys. Rev. E 56, 2702 (1997); H. Kallabis
and J. Krug, Europhys. Lett. 45, 20 (1999); A. J. Bray, S. N.
Majumdar, and G. Schehr, Adv. Phys. 62, 225 (2013).
[40] M. Prähofer and H. Spohn, J. Stat. Phys. 115, 255 (2004).
[41] Theoretical curves available in the following URL wereused: https://www-m5.ma.tum.de/KPZ.
[42] T. Halpin-Healy, Phys. Rev. Lett. 109, 170602 (2012); Phys.
Rev. E 88, 042118 (2013); T. J. Oliveira, S. G. Alves, andS. C. Ferreira, Phys. Rev. E 87, 040102(R) (2013).

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