This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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Initial perturbation matters: implications of geometry-dependent
universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
Yohsuke T. Fukai1, 2 and Kazumasa A. Takeuchi2
1) 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
2) Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan(*Electronic mail: ysk@yfukai.net)(Dated: 10 October 2021)

Infinitesimal perturbations in various systems showing spatiotemporal chaos (STC) evolve following the power laws of
the Kardar-Parisi-Zhang (KPZ) universality class. While universal properties beyond the power-law exponents, suchas distributions and correlations, and their geometry dependence, are established for random growth and related KPZ
systems, the validity of these findings to deterministic chaotic perturbations is unknown. Here we fill this gap between
stochastic KPZ systems and deterministic STC perturbations by conducting extensive simulations of a prototypical STC
system, namely the logistic coupled map lattice. We show that the perturbation interfaces, defined by the logarithm of
the modulus of the perturbation vector components, exhibit the universal, geometry-dependent statistical laws of the
KPZ class, despite the deterministic nature of STC. We demonstrate that KPZ statistics for three established geometries
arise for different initial profiles of the perturbation, namely point (local), uniform, and “pseudo-stationary” initial perturbations, the last being the statistically stationary state of KPZ interfaces given independently of the Lyapunov vector.
This geometry dependence lasts until the KPZ correlation length becomes comparable to the system size. Thereafter,
perturbation vectors converge to the unique Lyapunov vector, showing characteristic meandering, coalescence, andannihilation of borders of piece-wise regions that remain different from the Lyapunov vector. Our work implies that
the KPZ universality for stochastic systems generally characterizes deterministic STC perturbations, providing new
insights for STC, such as the universal dependence on initial perturbation and beyond.
Many studies have been devoted to elucidating universal properties of spatially extended dynamical systems
via statistical physics. In particular, seminal work by
Pikovsky, Kurths, and Politi has demonstrated that infinitesimal perturbations to various dynamical systems
showing spatiotemporal chaos (STC) evolve following universal power laws of the Kardar-Parisi-Zhang (KPZ)
universality class, a representative universality class forstochastic processes1,2 . Though exact studies on stochastic interface growth have advanced the knowledge of the
KPZ class dramatically3–9 , revealing geometry-dependent
but universal statistical properties beyond the power-law
exponents, these developments have been attained independently from the field of dynamical systems. In thispaper, we attempt to bridge those two subjects by extensive numerical simulations of the logistic coupled map lattice, a prototypical model of spatially extended dynamicalsystems10,11 . We first show clear evidence that the perturbations show the universal statistical properties of the
KPZ class beyond the power-law exponents, and the geometry dependence manifests itself as the dependence on
the initial profile of the perturbations. We then study thetime regime in which the KPZ correlation length reaches
the system size and found that the perturbations converge
to a unique Lyapunov vector as suggested by Oseledec’s
theorem12–14 . We found that this process is governed bycharacteristic wandering and annihilation of the boundaries of the segmental regions that differ from the Lyapunov vector. With the current knowledge on KPZ anddynamical systems, we expect our results to generally holdin STC and can be extended to general initial perturba-

tions.

I.

INTRODUCTION

To understand general laws governing spatially extended
dynamical systems, it is insightful to focus on the thermodynamic limit. Successful approaches include applying theconcept of phase transition to understand collective states15–19
and synchronization20 in large dynamical systems. Considering the infinite-size limit is also crucial for defining the extensivity of chaos, by the existence of a finite density of Lyapunov exponents per degree of freedom21–23 .
In particular, universality associated with scale invariance
has successfully provided a general insight for growing infinitesimal perturbations in spatially extended dynamical systems. When a system is showing spatiotemporal chaos(STC), i.e., chaos characterized by spatial and temporal disorder, perturbations given to trajectories typically grow exponentially and fluctuate. Pioneering studies by Pikovsky,
Kurths, and Politi1,2 revealed that, for a wide range of systems, this growing perturbation belongs to the Kardar-ParisiZhang (KPZ) universality class24 , which is one of the mostwide-ranging universality classes for various stochastic processes (see reviews3–9 ), including growing interfaces, directed
polymers, stochastic particle transport, nonlinear fluctuating hydrodynamics8,25 , and most recently, to quantum spin
chains26–29 , to name but a few.
To be more specific, let us focus on a system showing STC
in one-dimensional space. The dynamical variable is u(x,t)

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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2with position x and time t. We consider an infinitesimal
perturbation δ u(x,t), growing from a given initial perturbation δ u(x, 0) applied to the trajectory. Then, Pikovsky andcoworkers1,2 studied fluctuations of h(x,t) := log |δ u(x,t)|,which we call the perturbation interface, and found universal
power laws for the fluctuation amplitude ∼ t β and the correlation length ∼ t 1/z , with characteristic exponents β = 1/3 andz = 3/2 of the one-dimensional KPZ class. This emergence
of KPZ fluctuations in STC perturbations has been foundin a variety of systems, ranging from coupled map lattices(CML)1,2,30–32 , a coupled ordinary differential equation31,32 ,partial differential equations2 to time-delayed systems2,32,33 .
However, this connection to the KPZ class has far-reaching
implications, beyond the exponents. The understanding on
the one-dimensional KPZ class has been completely renewed
during the last decades, thanks to remarkable developments
by exact solutions3–9 . An important outcome is the determination of the distribution and correlation functions of the interface fluctuations, which turned out to be universal yet dependent on the initial condition3–6,9 . More specifically, we
can write down the interface height h(x,t) for infinite-size systems, asymptotically, in the following form:h(x,t) ' v∞t + (Γt)1/3 χ(X,t),

FIG. 1. Typical snapshots of h(x,t) = log |δ u(x,t)| from the uniform(left) and point (right) initial perturbations, at t = 0, 20, · · · , 100 frombottom to top.

fact? In this article, we address these questions numericallyby an extensive simulation of the CML of the logistic map, aprototypical model showing STC10,11 .
II.

METHODWe employed the CML of the logistic map, defined asfollows. The dynamical variables u(x,t) ∈ [0, 1] are definedfor integer x and t, with the periodic boundary condition
u(x,t) = u(mod(x, L),t). Time evolution is(1)

where χ(X,t) is a stochastic variable discussed below,
X := x/ξ (t) is the coordinate rescaled by the correlation
length ξ (t) := A2 (Γt)2/3 , and v∞ , Γ, A are system-dependent
parameters9,34 . Then, it has been shown both theoretically3,4,9and experimentally9,35–40 that statistical properties of χ(X,t)
are universal but depend on the choice of the initial condition h(x, 0). The following constitute three canonical cases,
dubbed universality subclasses: (i) the flat subclass for theflat initial condition h(x, 0) = 0; (ii) the circular subclass forwedge or curved initial conditions, e.g., h(x, 0) = −κ|x| or−cx2 ; (iii) the stationary subclass for statistically stationary
initial conditions. Note that√ such initial conditions typicallytake the form of h(x, 0) = AB(x), where B(x) is the standard
Brownian motion characterized by h[B(x + `) − B(x)]2 i = `and B(0) = 0. This is true exactly of the KPZ equation9,24,41
and also valid for other models in coarse-grained scales. Then,for each subclass, the one-point distribution of χ(0,t → ∞)is known to be (i) the Gaussian orthogonal ensemble (GOE)
Tracy-Widom distribution42 , (ii) the Gaussian unitary ensemble (GUE) Tracy-Widom distribution43 , and (iii) the BaikRains distribution44 (see3,4,6,9,45 ), denoted by the random variable χ1 , χ2 , χ0 , respectively46 . These subclasses have also
been characterized by correlation functions3–9 , which continue attracting considerable attention (e.g., recent developments on time covariance37,47–51 and on general initialconditions52,53 ).
With this background, the following questions arise: Does
this initial-condition dependence of KPZ fluctuations show up
in perturbation dynamics of STC? If yes, how? Moreover,since we expect that the vector direction of δ u(x,t) eventually converges to the unique Lyapunov vector associated withthe largest Lyapunov exponent, as Oselsedec’s theorem12–14
suggests unless the largest exponent is degenerate, how does
the initial-condition dependence of δ u(x,t) reconcile with thisu(x,t + 1) =(1 − ε) f (u(x,t))ε+ [ f (u(x − 1,t)) + f (u(x + 1,t))],2(2)which is denoted by Fx (u(0,t), . . . , u(L − 1,t)), with f (u) =au(1 − u) and parameters ε and a.
Here we chosea = 4 and ε = 0.05, for which this CML shows
fully disordered STC10,11 .
The evolution of infinitesimal perturbations δ u(x,t) is then given by δ u(x,t + 1) =
∑L−1
x0 =0

∂ Fx (u0 ,...,uL−1 )δ u(x0 ,t).
∂ ux0
u =u(i,t)i

To discard the transient before we applied a perturbation at
t = 0, each simulation started at t = −Twarmup from an initial condition u(x, −Twarmup ) generated randomly and independently. We chose Twarmup = 1000 and checked that thesystem is already in the attractor at t = 0, by comparing thehistograms of u(x,t) at t = 0 and t = 105 (Fig. S154 ). We thenapplied a perturbation δ u(x, 0) that takes different forms as we
describe below, obtained δ u(x,t), and analyzed fluctuations ofh(x,t) = log |δ u(x,t)|.
III.

UNIFORM AND POINT INITIAL PERTURBATIONS

We first studied the initial-shape dependence of STC perturbations. For this, we compare the uniform initial condition
δ u(x, 0) = 1 and the point initial condition δ u(x, 0) = δx,0 .
Figure 1 shows typical snapshots of the perturbation interface
h(x,t) = log |δ u(x,t)| for those initial perturbations. We simulated 2400 and 160000 independent trajectories with L = 215
and L = 212 for the uniform and point perturbations, respectively.
First, we measured the mean height hh(0,t)i, where h. . . i
indicates the ensemble average. For the uniform perturbations, we also took the spatial average because of theTo characterize statistical properties further, we estimatedthe nonuniversal parameters, v∞ , Γ, and A9,55 . These
parameters are expected to be independent of the initial
perturbation9,37,55 . First we estimated the asymptotic growth
speed v∞ , which corresponds to the Lyapunov exponent in thelimit L → ∞. To do so, we used the local Lyapunov exponent λ (t) := h ∂ h(0,t)∂t i, which satisfies, for sufficiently large
systems, the following relation derived from Eq. (1):λ (t) = v∞ + const. × t −2/3 .

102∝ t 2/3

101

uniform
point

103 t

(c)
λ∞ (L)

0.525

105

0
0.00

0.53250.53220.005 L0.01 t −2/3 0.02
−1

0.3

uniform skewness
point skewness
uniform kurtosis
point kurtosis

Sk [χ1 ]Sk [χ2 ]0.2

Ku [χ1 ]0.1

Ku [χ2 ]102t(d) 100
uniform
point

0.530

Sk [h], Ku [h](b)probability density

We then investigated the one-point distribution of h(0,t),measuring the skewness Sk[h(0,t)] := hh(0,t)3 ic/hh(0,t)2 i3/2
andcthe kurtosis Ku[h(0,t)] := hh(0,t)4 ic/hh(0,t)2 i2c . Figure 2(b) showsthat those for the uniform and point initial perturbation converge to the values for the flat (χ1 ) and circular (χ2 ) KPZ subclasses, respectively. This indicates that those perturbationsindeed belong to the respective KPZ subclasses.

(a)
hh2ic

translational symmetry. As expected, hh(0,t)i grows linearly with time, reflecting the exponential growth of theperturbation (Fig. S254 ). We also measured the variancehh(0,t)2 ic = hδ h(0,t)2 i with δ h(x,t) := h(x,t) − hh(x,t)i andthe third- and fourth-order cumulants, hh(0,t)3 ic = hδ h(0,t)3 i
and hh(0,t)4 ic = hδ h(0,t)4 i − 3hδ h(0,t)2 i2 . The kth-order
cumulants (k = 2, 3, 4) show power laws hh(0,t)k ic ∼ t k/3
characteristic of the KPZ class [Figs. 2(a) and S254 ], indicating that this exponent does not depend on the choice of initial
perturbation, as expected.

λ (t)This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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3

104

10−1
10−2
−4

uniformpoint−2 q 02

FIG. 2. Results for the uniform and point initial perturbations. (a)The variance of the height h(0,t). The black line is a guide to eyes indicating the power law t 2/3 . (b) The skewness Sk[h(0,t)] and the kurtosis Ku[h(0,t)] of the height distribution. The values for χ1 and χ2
are shown by the dashed and dotted lines, respectively. (c) The local
Lyapunov exponent λ (t) plotted against t −2/3 . The black lines are
the results of the linear fit. (inset) The infinite-time Lyapunov exponent λ∞ (L) plotted against L−1 . The black line is the result of the linear fit used for the estimation of Aλ̃ and v∞ . (d) The rescaled height
distribution at t = 103 (lighter color) and t = 105 (darker color), for
the uniform (turquoise squares) and point (red triangles) perturbations. The probability density function of χ1 and χ2 (GOE and GUE
Tracy-Widom distributions, respectively) is shown by the solid anddashed lines, respectively.

(3)
With those parameters, we defined the rescaled height

Using this, we estimated v∞ by linear regression of λ (t)against t −2/3 [Fig. 2(c)].
We thereby obtained v∞ =0.53249(2) and 0.53250(7) for the uniform and point initial
perturbations, respectively, where the numbers in the parentheses indicate the estimation uncertainty. This confirms thatv∞ does not depend on the initial perturbation. Next, we estimated Γ by hh(0,t)2 ic/(hχ 2j ic t 2/3 ) ' Γ2/3 derived from Eq. (1),using χ1 and χ2 for the uniform and point initial perturbations, respectively. Similarly to other KPZ interfaces9,37,56 ,the quantity in the left-hand side shows finite-time corrections in the order of O(t −2/3 ) (Fig. S354 ). Therefore, by
linear regression against t −2/3 , we obtained Γ = 0.1373(5)and 0.1369(6) for the uniform and point initial perturbations, respectively. From this, we derived the final estimate
Γ = 0.1371(9). Finally, we estimated A from the asymptotic growth speed for the finite-size systems, or the Lyapunov exponent, λ∞ (L) := limt→∞ h ∂∂th i. First, note that Eq. (3)for λ (t) is expected to hold until the correlation length ξ (t)reaches the system size L. Beyond this, λ (t) will deviate from
Eq. (3) and converge to λ∞ (L) ' v∞ − ALλ̃ , where λ̃ is a paA2 |λ̃ |rameter satisfying Γ = 2 9,55 . Our simulations for the uniform initial perturbation (see Table SI54 for parameter values)
confirmed that λ∞ (L) depends linearly on L−1 [Fig. 2(c) inset]. By linear regression, we obtained Aλ̃ = 0.0744(3) andv∞ = 0.5324985(7), the latter refining our previous estimates.
From the estimates of Aλ̃ and Γ, we obtained A by A = A2Γλ̃ .

q(0,t) :=h(0,t) − v∞t' χ(0,t)(Γt)1/3(4)and investigated its statistical properties. Figure 2(d) showsits one-point distribution (again taking all X for the uniform
perturbation). The distributions for the uniform and point initial perturbation agree with those of the exact solutions for theflat and circular subclasses, namely the GOE and GUE TracyWidom distributions, respectively. Accordingly, the first- tofourth-order cumulants of q(0,t) converge to those of χ1 andχ2 , respectively (Fig. S454 ). We also measured spatial andtemporal correlation functions and found agreement with the
known results for the flat and circular subclasses (see Supplemental Text54 ). These results clearly indicate that the statistical properties of perturbation dynamics for the uniform andpoint initial perturbation are governed by the flat and circular
KPZ subclasses, respectively.
IV.

PSEUDO-STATIONARY INITIAL PERTURBATION

From the viewpoint of dynamical systems theory, the perturbation interfaces are expected to converge to the unique
Lyapunov vector (unique at each point in phase space), associated with the largest Lyapunov exponent13,14 . On the other
hand, KPZ interfaces generally have a statistically stationary
state, characterized by the time-invariance of the equal-timeCh (∆x,t) := h[h(x + ∆x,t) − h(x,t)] i ' A∆x,

(5)which generally holds in the stationary state of the onedimensional KPZ class3–9,41 . However, since the notion ofBrownian motion can be defined only by the ensemble of realizations, for STC perturbations, it does not specify the uniquerealization corresponding to the Lyapunov vector. In otherwords, all realizations of the Brownian motion except the one
corresponding to the true Lyapunov vector are not the stationary state of the perturbation interfaces. It is therefore natural
to ask whether such pseudo-stationary states, generated randomly irrespective of the state of the dynamical system, may
exhibit the characteristic fluctuation properties of the stationary KPZ subclass.
To answer this, we studied perturbation interfaces frompseudo-stationary initial conditions, given independently of
the state of the dynamical system. Using the estimated valueof A, we√ prepared a pseudo-stationary initial perturbation
h0 (x) = AB(x) with h0 (0) = 0 and the periodic boundary
condition (see Supplemental Text54 ), and set the initial perturbation by δ u(x, 0) = exp[h0 (x)]. Figure 3(a) illustrates
the perturbation interface for such a pseudo-stationary initialcondition. We simulated 2400 sets of independent trajectories and pseudo-stationary initial perturbations, with L = 215 .
To evaluate√ systematic error due to the uncertainty of A inh0 (x) = AB(x), we also ran simulations with the value ofA replaced with its upper and lower bounds of the range of
uncertainty, 2400 realizations each.
We first checked the statistical stationarity of perturbationinterfaces from such pseudo-stationary initial conditions. Figure 3(b) shows that Ch (∆x,t)/∆x at t = 104 and 105 are identical in the pseudo-stationary case, keeping Ch (∆x,t)/∆x ' A
for ∆x  L. Figure 3(c) shows that the growth speed, or the
local Lyapunov exponent, takes a constant value only aftera very short period. These results demonstrate the statisticalstationarity of our pseudo-stationary interfaces.
We then compared the statistical properties of the perturbation interfaces with the predictions for the KPZ stationary
subclass. In the following, the statistical accuracy was improved by taking advantage of translational symmetry, whichguarantees that h(x,t) − h0 (x) is equivalent to h(0,t). We firstmeasured the skewness and kurtosis of h(0,t) and found convergence to those of χ0 , i.e., the Baik-Rains distribution characterizing the stationary subclass [Fig. 3(d) inset]. We also
confirmed agreement with the Baik-Rains distribution, by histograms [Fig. 3(e)] and cumulants (Fig. S454 ) of the rescaledheight q(0,t). Agreement with the stationary KPZ subclass
was also confirmed through correlation functions (see Supplemental Text54 ), notably by quantitative agreement to the exact solution of the two-point space-time correlation57 . These
results demonstrate that the perturbation interfaces from thepseudo stationary initial conditions are indeed governed by
the stationary KPZ subclass, despite not being stationary fort = 40t = 60t = 80t = 100

0

(c)

−20

0

x

20

0.53

0.52

102

uniform
point
pseudo stationary
104tCh (∆x , t)/∆x

(b)

50

100
10−1
10−2
101

(d) 100
probability density2(a)h(x , t) = log |δu|probability distribution of the height fluctuations. For the KPZ
equation, such a stationary√ state is given by the Brownian
motion9,24,41 , h(x, ·) = AB(x), as already described. Thisis characterized by the scaling law for the height-difference
correlation functionλ (t)This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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4
t=0
t = 20

10

−1

10

−2

uniform
pseudo stationary

∆x 10

3

Sk [h], Ku [h]0.4

Sk [χ0 ]0.3

Ku [χ0 ]102 t 104
pseudo stationary−2.5
0.0 q 2.5

skewnesskurtosis

5.0

FIG. 3.
Results for the pseudo-stationary initial perturbations.
(a) Typical snapshots of perturbation interfaces at t = 0, 20, · · · , 100from bottom to top. (b) The height-difference correlation function
Ch (∆x,t) divided by ∆x at t = 104 (lighter color) and t = 105 (darker
color). The dashed line shows the value of A. (c) The local Lyapunov
exponent plotted against time. (d) The height distribution at t = 103(lighter color) and t = 105 (darker color). The solid line shows the
Baik-Rains distribution. (inset) The skewness and the kurtosis ofh(x,t). The shaded area indicatesthe systematic error due to the un√certainty of A in h0 (x) = AB(x) (see text). The dash-dotted lines
show the values for the Baik-Rains distribution.

the dynamical system. Of course, one can also start with thetrue Lyapunov vector as an initial perturbation and the same
KPZ stationary subclass is expected.

V. CONVERGENCE TO THE LYAPUNOV VECTOR IN
FINITE-SIZE SYSTEMS

We have shown so far the initial-shape dependence of perturbation vectors, focusing on the three cases corresponding to
the KPZ subclasses. On the other hand, for finite-size systems,
we expect that the direction of the perturbation vectors eventually converges to that of the unique Lyapunov vector13,14 ,whence the initial-condition dependence is lost. This is expected to happen when the correlation length ξ (t) reaches thesystem size L. Here we investigated how this happens forthe three studied initial perturbations, using the same system
but with L = 210 . With this size, the correlation length ξ (t)3/2 ≈ 6 × 105 .
reaches L at t ∗ := Γ−1 ( AL2 )To obtain the Lyapunov vector, we started simulations fromt = −Twarmup − t0 with t0 := 6 × 106 ( t ∗ ). After discarding
the trajectory’s transient for Twarmup time steps, we applieda uniform perturbation δ u0 (x, −t0 ) = 1 and let it evolve fort0 time steps. Then we defined hstat (x,t) := log |δ u0 (x,t)| −log |δ u0 (0, 0)|, which is the interface corresponding to the
Lyapunov vector and we call it the stationary perturbation interface. We compared it with the uniform, point, and pseudostationary interfaces h(x,t) obtained for the same trajectory,
and found that these indeed converge to hstat (x,t) up to a
constant shift which disappears by normalization of δ u(x,t)[Fig. 4(a)]. Interestingly, before this convergence, the difference h(x,t) − hstat (x,t) consists of piece-wise plateaus, rem-

uniform

(a)

×103
1.1

50
0
50
0

h
h hstatThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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5

point

×103
8.55
8.50
8.45
0

1.0
0

pseudo stationary
×104

6.71

stationary

×102+5.32 × 105
5
4

6.70
0

0

100
100
100
100
100
500 0 500 500 0 500 500 0 500 500 0 500 500 0 500(b) (×105)
2

x

t=0
uniform

t1
0500

x0

x

x

x

x

t = 2 × 103 t = 1.6 × 104 t = 1.3 × 105 t = 1 × 106
h hstat (×105) point
2
0
20
t1
40
0500 0500x

pseudoh hstat (×105) stationary h hstat
200
50 t 110100
200500 0500
500xFIG. 4. Convergence of the perturbation interfaces to the Lyapunov
vector. (a) Typical snapshots, taken at time indicated below eachpanel, are shown. (b) Typical evolution of h(x,t) − hstat (x,t).

iniscent of the difference between the first and subsequent
Lyapunov vectors31 . The plateau edges meander in an apparently random manner, coalesce, and eventually annihilate[Fig. 4(b)].

VI.

Recent studies have shown that the concept of the KPZ
subclasses is also useful for understanding deterministic
fluctuations of STC itself, in the Kuramoto-Sivashinsky
equation61,62 , in anharmonic chains8,25,63 , in a discrete GrossPitaevskii equation64 , etc. We hope that this line of research
will continue providing new insights for spatially extended
dynamical systems.

SUMMARY AND DISCUSSIONIn summary, using the logistic map CML as a prototypicalmodel showing STC, we studied the distribution and correlation properties of perturbation vectors, for the following three
types of initial perturbations: uniform, point, and pseudostationary. Focusing on the distribution and correlation properties, we showed clear evidence of agreement with the flat,
circular, and stationary subclasses of the KPZ class, respectively. Since the KPZ exponents have been reported for perturbations in various STC systems1,2,32 , it is reasonable to believe
the generality of our results. In the pseudo-stationary case, thestationary KPZ subclass arises despite the fact that such a perturbation was not the Lyapunov vector and therefore not stationary for the dynamical system. These results indicate thatthe deterministic nature of dynamical systems does not affectthe universal statistical properties of the KPZ class. When thecorrelation length eventually reaches the system size, the perturbation interfaces from different initial perturbations converged to the unique stationary interface up to constant shifts,
unlike stochastic interface growth. This convergence is driven
by intricate meandering, coalescence, and eventual annihilation of plateau borders seen in the height difference from thestationary interface.
Although we studied only the three types of initial perturbations here, the KPZ theory can also deal with general initial perturbations, via the KPZ variational formula40,58–60 . We
expect that we can describe the one-point distribution of perturbation vectors for arbitrary initial conditions, by using thenumerical method proposed in40 and by estimating nonuniversal parameters according to the procedure described in thisarticle.

SUPPLEMENTARY MATERIAL

See supplementary material54 for Supplemental Texts onuniversal correlation functions and pseudo-stationary initialconditions, Table SI and Figs. S1, S2, S3, S4, S5 and S6.

ACKNOWLEDGMENTS

We thank K. Kawaguchi and S. Ito for useful discussions on
the Brownian initial condition with periodic boundary conditions. We are grateful to M. Prähofer and H. Spohn for thetheoretical curves of the Baik-Rains and Tracy-Widom distributions and that of the stationary two-point correlation function, which are made available online65 , and to F. Bornemann
for the curves of the Airy1 and Airy2 covariance, evaluated byhis algorithm in66 . This work is supported in part by KAKENHI from Japan Society for the Promotion of Science (Grant
Nos. JP17J05559, JP25103004, JP19H05800, JP20H01826).

DATA AVAILABILITY STATEMENT

The data that support the findings of
study
are
openlyavailableinZenodohttps://doi.org/10.5281/zenodo.5559231.

this
at

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This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0071658

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