Nonequilibrium criticality driven by Kardar-Parisi-Zhang uctuations in the synchronization of oscillator lattices Ricardo Guti errez1and Rodolfo Cuerno1

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Nonequilibrium criticality driven by Kardar-Parisi-Zhang ﬂuctuations in the

synchronization of oscillator lattices

Ricardo Guti´errez1and Rodolfo Cuerno1

1Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matem´aticas,

Universidad Carlos III de Madrid, 28911 Legan´es, Madrid, Spain

The synchronization of oscillator ensembles is pervasive throughout nonlinear science, from clas-

sical or quantum mechanics to biology, to human assemblies. Traditionally, the main focus has been

the identiﬁcation of threshold parameter values for the transition to synchronization as well as the

nature of such transition. Here, we show that considering an oscillator lattice as a discrete growing

interface provides unique insights into the dynamical process whereby the lattice reaches synchro-

nization for long times. Working on a generalization of the celebrated Kuramoto model that allows

for odd or non-odd couplings, we elucidate synchronization of oscillator lattices as an instance of

generic scale invariance, whereby the system displays space-time criticality, largely irrespective of

parameter values. The critical properties of the system (like scaling exponent values and the dy-

namic scaling Ansatz which is satisﬁed) happen to fall into universality classes of kinetically rough

interfaces with columnar disorder, namely, those of the Edwards-Wilkinson (equivalently, the Larkin

model of an elastic interface in a random medium) or the Kardar-Parisi-Zhang (KPZ) equations,

for Kuramoto (odd) coupling and generic (non-odd) couplings, respectively. From the point of view

of kinetic roughening, the critical properties we ﬁnd turn out to be quite innovative, especially con-

cerning the statistics of the ﬂuctuations as characterized by their probability distribution function

(PDF) and covariance. While the latter happens to be that of the Larkin model irrespective of

the symmetry of the coupling, in the generic non-odd coupling case the PDF turns out to be the

Tracy-Widom distribution associated with the KPZ nonlinearity. This brings the synchronization

of oscillator lattices into a remarkably large class of strongly-correlated, low-dimensional (classical

and quantum) systems with strong universal ﬂuctuations.

I. INTRODUCTION

From chirping crickets to Josephson junctions and

quantum oscillators, passing through cells in the heart

and in the brain, a huge variety of systems across all of

science exhibit synchronous dynamics [1, 2]. While the

scientiﬁc study of synchronization can be traced back in

time to the work of Christiaan Huygens in the 17th cen-

tury, it is in the last few decades that it has become

a central concept in nonlinear and complex dynamical

systems, as a pervasive form of emerging collective dy-

namics. It is frequently studied using models of phase

oscillators (i.e. idealized limit-cycle oscillators), as in the

well-known Kuramoto model [3, 4], though in fact it has

been studied in low-dimensional chaotic systems as well

[5], even in complex networks of such systems [6].

Another seemingly unrelated subject that has focused

a great deal of attention in the last few decades is the

study of surface kinetic roughening [7, 8], which also uni-

ﬁes a great diversity of nonequilibrium phenomena, from

the production of thin solid ﬁlms to the growth of bac-

terial colonies, or the formation of coﬀee rings [9]. The

ﬁne details of many experimental systems and theoretical

models in this context have been found to become irrel-

evant at suﬃciently large space and time scales, where

they show traits of universality akin to that of equilib-

rium critical dynamics [10, 11]. Crucially, however, now

space-time criticality does not require adjusting control

parameters to precise critical values but appears, rather,

over a region of parameter space with nonzero measure.

Thus, surface kinetic roughening constitutes an impor-

tant instance of generic scale invariance (GSI) [12–14], a

concept which is in turn closely related to that of self-

organized criticality [15, 16].

A key player in the GSI realm is the Kardar-Parisi-

Zhang (KPZ) stochastic equation [17–19], whose univer-

sality class is recently proving paradigmatic for the space-

time critical behavior of ﬂuctuations in low-dimensional,

strongly correlated systems. Examples range from non-

quantum systems like active matter [20], turbulent liq-

uid crystals [21], stochastic hydrodynamics [22], col-

loidal aggregation [23], thin-ﬁlm deposition [24], reaction-

diﬀusion processes [25], or random geometry [26], to the

quantum realm, including exciton polariton condensates

[27, 28] or quantum entanglement [29], integrable and

non-integrable quantum spin chains [30–33], or electronic

ﬂuids [34]. Probably, such a ubiquity for the KPZ univer-

sality is in turn related with the fact that the statistics

of ﬂuctuations is described [18, 19] by the Tracy-Widom

(TW) family of probability density functions (PDF) for

the maximum eigenvalue of Hermitian random matrices

[35, 36]. Indeed, TW statistics is recently being found

ubiquitously [37] across length scales in natural and tech-

nological systems, providing an analog of Gaussian statis-

tics for correlated variables.

Connections between oscillator synchronization and

surface kinetic roughening have been occasionally

pointed out in the literature —see e.g. Refs. [1, 38, 39]

and others therein—, often through mappings to the KPZ

equation, to its Gaussian approximation, the so-called

Edwards-Wilkinson (EW) equation [9, 40], or variations

thereof. For instance, a discretized KPZ equation has

arXiv:2210.03782v2 [cond-mat.stat-mech] 21 Apr 2023

been recently shown to approximate a noisy Kuramoto-

Sakaguchi model of an (one or two-dimensional) oscilla-

tor lattice [41], while the related compact KPZ equation

has been shown to describe the dynamics of the phase of

a driven-dissipative Bose-Einstein condensate of exciton

polaritons [42]. Even more recently, randomness in the

natural frequencies of the latter system has been shown

[43] not to destroy synchronization and KPZ scaling has

been experimentally reported in a 1D polariton conden-

sate [28]. In a related context, KPZ universality is known

to be relevant in the hydrodynamic phase ﬂuctuations

of spatially-extended nonequilibrium oscillating systems,

see e.g. Refs. [44, 45]. However, a systematic assessment

of the relevance and implications of KPZ ﬂuctuations to

the synchronization of oscillator lattices does not seem

available in the literature yet.

In this paper, we aim to ﬁll in this gap. Focusing on a

generalization of the model of one-dimensional (1D) os-

cillator lattices addressed in Refs. [41, 43] —which allows

for both odd (as in the seminal Kuramoto model [3, 46])

and non-odd coupling functions among oscillators—, we

unambiguously elucidate synchronization in these sys-

tems as an instance of GSI, with anomalous forms of

scaling and universal ﬂuctuation statistics which are most

conveniently phrased in terms of those characterizing GSI

in surface kinetic roughening, and in which KPZ ﬂuc-

tuations play an important role. The lack of awareness

that the dynamics of synchronizing oscillators is endowed

with universal features due to nonequilibrium criticality

probably explains why the dynamical process whereby

synchronization is achieved has remained poorly studied.

Previous analytical results [47] show that synchroniza-

tion of oscillator lattices is not possible in the thermo-

dynamic limit for odd coupling functions. Viewing the

oscillator array as an evolving interface, we show this lack

of synchronization to be a consequence of so-called super-

rough kinetic roughening [48], whereby the local slopes

along the interface only stabilize upon saturation [49, 50].

Moreover, the odd symmetry of the coupling function be-

comes the condition for which synchronization features

the precise scaling behavior of the EW model with so-

called columnar noise, known as the Larkin model in the

context of elastic interfaces in disordered media; see, e.g.,

Ref. [51] and references therein. For generic coupling

functions, on the other hand, we show that synchroniza-

tion becomes possible for arbitrarily large systems [52],

leading to the growth of faceted interfaces (oscillator clus-

ters) where the slopes stabilize earlier [53]. The large-

scale eﬀective description is now given by the KPZ equa-

tion with columnar noise, which however is not in the

universality class of the celebrated KPZ equation [54].

Diﬀerent synchronous dynamics thus map into diﬀerent

surface growth models.

From the stand point of kinetic roughening, the dy-

namic scaling Ansatz satisﬁed by the space-time ﬂuctu-

ations of the oscillator array is anomalous [48–50, 53],

akin to those found in many systems with morpholog-

ical instabilities and/or quenched disorder [55]. What

is more striking is that for non-odd coupling functions,

phase ﬂuctuations follow the ubiquitous TW PDF, one

of the hallmarks of KPZ universality, in spite of the fact

that neither the scaling Ansatz, nor the critical exponent

values, nor the covariance of the ﬂuctuations are those

of the 1D KPZ universality class. This is in line with

very recent observations in continuous [56] and discrete

[57] models that, somewhat unexpectedly, the ﬂuctua-

tion PDF and covariance, and the scaling exponents are

all independent traits of a GSI universality class.

This paper is organized as follows. In Section II we de-

scribe ﬁrst the connection between synchronization and

kinetic roughening, followed by a discussion of the ob-

servables of interest. Section III contains a description

of the particular model to be considered in the numerics

and a general discussion and results on the synchronous

dynamics. Section IV is devoted to the study of syn-

chronization with the Kuramoto (sine) coupling form,

for which we solve the continuum-limit (linear) Larkin

model. Numerical simulations reveal a super-rough scal-

ing adequately described by the analytical results. In

Section V we study synchronization with a non-odd cou-

pling function, whose eﬀective description yields a (non-

linear) KPZ equation with columnar noise. Sections IV

and V also contain the study of the one- and two-point

statistics of ﬂuctuations that yields Gaussian (respec-

tively, TW) statistics for odd (respectively, non-odd) cou-

plings, but the same covariance of the Larkin model in all

cases. Finally, in Section VI we provide some concluding

remarks and ideas for future work. Additional numerical

results are organized into ﬁve appendices at the end.

II. SYNCHRONIZATION VS KINETIC

ROUGHENING

A. General lattices of phase oscillators

We consider a system of Ldphase oscillators at the

sites of a d-dimensional hypercubic lattice of linear size

L. Each oscillator is an idealized dissipative dynamical

system with an attracting limit cycle, whose state is given

by a phase φi(t). The time evolution is determined by

its intrinsic frequency ωiand the interactions with its

neighbors through a smooth, 2π-periodic coupling func-

tion Γ(φj−φi),

dφi

dt =ωi+X

j∈Λi

Γ(φj−φi), i = 1,2, . . . , Ld,(1)

where Λiis the set of 2dneighbors of site i. The in-

trinsic frequencies are independent and identically dis-

tributed according to a probability density g(ω) with zero

mean and ﬁnite variance, i.e. hωii=Rdωg(ω)ω= 0 and

hωiωji=δij Rdωg(ω)ω2= 2σδij , where δij is the Kro-

necker delta. A nonzero mean would only introduce a

uniform frequency shift, easily removable by moving to

the rotating frame. Moreover, we assume that the dis-

tribution is even, g(−ω) = g(ω). The coupling function

Γ(φj−φi) is assumed to include a coupling strength K,

i.e. an overall nonnegative proportionality factor with

dimensions of inverse time.

The eﬀective frequencies of oscillation are deﬁned as

ωeﬀ

i≡lim

T→∞

φi(τ+T)−φi(τ)

T, i = 1,2, . . . , Ld,(2)

where [0, τ] is a time interval suﬃciently long to contain

the transient behavior, and the limit is assumed to exist.

Oscillators that evolve at the same eﬀective frequency are

said to be frequency entrained or frequency locked. The

kinetic-roughening observables that we consider focus on

this (time-averaged) form of synchronization, which does

not require the instantaneous frequencies dφi/dt to be-

come strictly identical at all times.

B. Continuum approximation

Generalizing the 1D approach of Ref. [46], we write the

positions of the oscillators as vectors in continuous space,

x= (x1, x2, . . . , xd)∈Rd, so the phase of oscillator i,φi,

is now denoted φ(x), and the neighboring oscillators are

placed at positions x±aek, where ekfor k= 1,2, . . . , d

is a canonical basis vector. Thus Eq. (1) becomes

∂tφ(x, t) = ω(x) +

k=1

[Γ(φ(x+aek, t)−φ(x, t))

+Γ(φ(x−aek, t)−φ(x, t))] ,(3)

where hω(x)i= 0, and hω(x)ω(x0)i= 2σδ(x−x0), us-

ing the Dirac delta. We will focus on a coarse-grained

description where ais assumed to be small compared to

the wavelengths over which the phase ﬁeld φ(x, t) ﬂuctu-

ates. Taylor-expanding the phase ﬁeld around x,

φ(x±aek, t)−φ(x) = ±a∂kφ(x, t) + 1

2a2∂2

kφ(x, t)

±1

6a3∂3

kφ(x, t) + O(a4),(4)

where ∂kis shorthand for the partial derivative with re-

spect to the k-th coordinate xk. In Eq. (3), after Taylor-

expanding the coupling functions, terms of odd order in

avanish due to the a→ −asymmetry of the coupling

term, yielding

∂tφ(x, t) = ω(x)+2dΓ(0) + a2Γ(1)(0)

k=1

∂2

kφ(x, t)

+a2Γ(2)(0)

k=1

(∂kφ(x, t))2+O(a4),(5)

where Γ(n)denotes the n-th derivative of Γ.

For a relatively slow spatial variation of the phase ﬁeld

φ(x, t), as occurs for coupling strengths Kwell into the

synchronized regime, it may be reasonable to neglect

terms of order higher than a2, which is the dominant

one for spatial coupling in the oscillating medium. By

analogy with surface growth, we will refer to this as a

small-slope approximation which yields the eﬀective con-

tinuum equation

∂tφ(x, t) = ω∗(x) + ν∇2φ(x, t) + λ

2[∇φ(x, t)]2,(6)

where ω∗(x)≡ω(x)+2dΓ(0), and as usual the Lapla-

cian ∇2φ(x, t)≡Pd

k=1∂2

kφ(x, t) and the squared norm

of the gradient [∇φ(x, t)]2≡Pd

k=1

[∂kφ(x, t)]2. The ap-

propriateness of the truncation for speciﬁc cases will be

discussed when making the comparison between predic-

tions derived from it and results based on the direct nu-

merical integration of particular instances of Eq. (1). A

similar continuum approximation for extended systems

with time-delayed couplings was analyzed and applied

to the study of long-wavelength modes of the vertebrate

segmentation clock in Ref. [58].

Equation (6) features the same deterministic derivative

terms as the KPZ equation [17]. We have introduced two

parameters, ν≡a2Γ(1)(0) and λ/2≡a2Γ(2)(0), following

the standard notation in the surface growth literature,

where they quantify smoothening surface tension and in-

terface growth along the local surface normal direction,

respectively [9]. In the case of the oscillators such names

at most provide an intuitive meaning to the parameters,

which are not necessarily positive. This formal connec-

tion between oscillator lattices and rough interfaces has

been pointed out on several occasions in the literature,

with some recent works even exploiting it for the study

of synchronization in novel scenarios [41, 43].

Notice, however, that in Eq. (6) the noise term, ω∗(x)

is time-independent, in contrast with the time-dependent

noise of the standard KPZ equation. In the presence of

such quenched disorder the system evolves deterministi-

cally from the initial condition. The dynamics is akin to

that of a growth process in a medium for which the dis-

order values depend on the substrate coordinate xbut

not on the local value of the interface “height” φ(x, t).

In the kinetic roughening literature, Eq. (6) is known as

the KPZ equation with columnar noise [7, 54]. Its linear

version, obtained for λ= 0, is the EW equation with

columnar noise, known also as the Larkin model.

C. Morphological analysis

In our analysis we consider the phase ﬁeld φ(x, t) as if

it were describing the height h(x, t) of an interface grow-

ing above point x∈Rdon a d-dimensional substrate, at

time t. In surface kinetic roughening processes [7–9], the

ﬂuctuations of the local height around the mean value

are captured by the global width or roughness

W(L, t)≡ h[h(x, t)−h]2i1/2,(7)

where the overbar denotes a spatial average in a system of

linear (substrate) size Land the angular brackets denote

averaging over diﬀerent noise realizations. GSI implies

that surface height values are statistically correlated for

distances smaller than a correlation length ξ(t) which

increases with time as a power law, ξ(t)∼t1/z, where

zis the so-called dynamic exponent. Such an increase

takes place until ξ(t) reaches a value comparable to L,

which results in the width saturating at a steady-state,

size-dependent value W(L, t Lz)∼Lα. Here, αis

the so-called roughness exponent, which is related with

the fractal dimension of the interface proﬁle h(x, t). In

a wide variety of physical contexts and conditions, the

global roughness satisﬁes the Family-Vicsek (FV) scaling

Ansatz [7–9, 59]

W(L, t) = tβf(L/ξ(t)),(8)

where the scaling function f(y)∼yαfor y1, while

f(y) reaches a constant value for y1. The ratio

β=α/z is known as the growth exponent, and character-

izes the short-time behavior of the roughness, W(t)∼tβ.

Notably, the FV Ansatz is veriﬁed by classical models of

equilibrium critical dynamics [11]. Away from equilib-

rium, it is also veriﬁed by representatives of important

universality classes of kinetic roughening, like those of the

KPZ and EW equations, which are characterized by the

set of (α, z) exponent values and their dependence on the

substrate dimension d[7–9]. In this sense, kinetic rough-

ening extends classical equilibrium critical dynamics far

from equilibrium [11].

Beyond global quantities like W(t), the GSI occurring

in kinetic roughening systems also impacts the behavior

of correlation functions. A particularly useful one is the

height-diﬀerence correlation function,

G(r, t)≡ h[h(x+r, t)−h(x, t)]2i.(9)

In our cases of interest, due to rotational invariance, the

correlations only depend on `≡ |r|. For FV scaling

G(`, t) scales diﬀerently depending on how `compares

with the correlation length [7–9],

G(`, t)∼t2β,if t1/z `,

`2α,if `t1/z.=`2αg(`/ξ(t)),(10)

where we are assuming ` < L and g(y) is a suitable scal-

ing function [8, 9]. In fact, G(`, t) scales like the square

of the local width w(`, t)≡ h[h(x, t)−h]2i1/2(the spa-

tial average is here restricted to a region of linear size

`), G(`, t)∼w2(`, t). One can see that the scaling form

in Eq. (10) only reﬂects the growth and saturation dis-

cussed above for the whole system, but now restricted to

a region of linear size `[60]. A related correlation func-

tion that is also frequently studied [19] is the so-called

height covariance

C(r, t)≡ hh(x, t)h(x+r, t)i−h¯

h(t)i2,(11)

such that, again under the assumption of rotational in-

variance, G(`, t) = 2[W2(t)−C(`, t)] [8].

Whenever the roughness exponent α≥1, it is useful

to consider [60, 61] an alternative two-point correlation

function, namely, the surface structure factor, i.e. the

power spectral density of the height ﬂuctuations, deﬁned

S(k, t)≡ hˆ

h(k, t)ˆ

h(−k, t)i=h|ˆ

h(k, t)|2i,(12)

where ˆ

h(k, t)≡ F[h(x, t)] is the space Fourier transform

of h(x, t) and kis d-dimensional wave vector. The FV

Ansatz now reads

S(k, t) = k−(2α+d)sFV(kt1/z),(13)

with sFV(y) approaching a constant value for y1 and

sFV(y)∝y2α+dfor y1. This can be derived by realiz-

ing that W2(L, t) equals the integral of S(k, t) over wave

vector space (Parseval’s theorem) [9]. Likewise, S(k, t) is

analytically related with, e.g., G(r, t) via space Fourier

transforms [8].

In the case of a system of oscillators, analogous observ-

ables to the global roughness and the correlation func-

tions, to be denoted as Wφ(L, t), Gφ(r, t), Cφ(r, t) and

Sφ(k, t), are simply deﬁned by replacing the height ﬁeld

h(x, t) by the phase ﬁeld φ(x, t). They will be the main

objects of our analysis in the sections to follow. Regard-

ing two-point correlations, we will be particularly inter-

ested in their value at a distance of one site, Gφ(`= 1, t),

which will be referred to as the average squared slope, and

denoted as h(∆φ)2i. Our interest lies in ﬁnite systems,

where saturation may be eventually attained. The key

point is that diﬀerences between oscillator phases that

do not evolve at the same eﬀective frequency ωeﬀ must

grow steadily in time for long times. Thus the phases

of two oscillators with eﬀective frequencies ωeﬀ

1and ωeﬀ

eventually separate, φ2(t)−φ1(t)∼(ωeﬀ

2−ωeﬀ

1)t. For this

reason, the saturation of Wφ(L, t) [or equivalently, that

of Sφ(k, t) or Gφ(r, t)] as t→ ∞, which shows that the

phase diﬀerences stop growing at some time, indicates

the presence of synchronization in the sense mentioned

above.

D. Anomalous scaling and universal ﬂuctuations in

the synchronization process

Two further aspects of surface growth, which have been

the focus of much recent research, will turn out to be in-

dispensable for the analysis of the synchronization prob-

lem. They both challenge the traditional characteriza-

tion of kinetic-roughening universality classes in terms

of just two independent exponents appearing in the FV

dynamic scaling Ansatz, Eq. (8). One is the existence

of growth processes with anomalous scaling properties

[8, 48–50, 53, 55]. While for standard FV systems height

ﬂuctuations at local distances `Lscale with the same

roughness exponent as global ﬂuctuations do at distances

comparable with the system size L, in systems displaying

anomalous scaling local and global ﬂuctuations scale with

diﬀerent roughness exponents, i.e. w(`, t `z)∼`αloc

with αloc 6=α.

The anomalous scaling that occurs in our work is most

conveniently identiﬁed by means of the structure factor,

as a new independent exponent αsappears (termed spec-

tral roughness exponent), in the dominant contribution in

Fourier space, namely [53],

S(k, t) = k−(2α+d)s(kt1/z),(14)

where s(y)∝y2(α−αs)for y1 and s(y)∝y2α+dfor

y1. Equation (14) generalizes the FV Ansatz, Eq.

(13), which is retrieved if αs=α. In general, the value

of αsand its relation with respect to αdetermines the

type of anomalous scaling [53].

We will be interested in cases such that αs>1, for

which the correlation function scales as [60]

G(`, t)∼t2β,if t1/z `L,

`2αloc t2(α−αlo c )/z,if `t1/z L. (15)

This means that the two-point correlations keep increas-

ing (anomalously) with time even at distances which are

smaller than the correlation length, at which they satu-

rate in case of FV scaling; in contrast, now they only sat-

urate at `2αloc L2(α−αlo c )when t1/z ∼L. If α=αs>1,

the anomalous scaling is termed super-rough [50], due to

the large interface ﬂuctuations that occur. The scaling

Ansatz satisﬁed by the structure factor is FV in this case,

but αloc = 1 6=α. Otherwise, if α6=αswith both ex-

ponents being larger than 1, faceted anomalous scaling

takes place [53], and again αloc = 1. There exist cases

(like that of the tensionless KPZ equation [62]) in which

αs<1, leading to so-called intrinsic anomalous scaling

for which αloc need not equal 1 [50].

A second aspect of surface kinetic roughening that is

relevant to our analysis has to do with the statistics of

ﬂuctuations. For rough surfaces, the observable of inter-

est is the PDF of the ﬂuctuations of the heights h(x, t) [in

our case, that of the phases φ(x, t)], around their mean.

By a straightforward adaptation of the deﬁnition em-

ployed in the kinetic roughening literature [7, 18, 19],

we will focus on the PDF of

ϕi≡δφi(t0+ ∆t)−δφi(t0)

(∆t)β,(16)

where δφi(t) = φi(t)−φ(t), t0is a reference time beyond

the initial transient dynamics, and t0+ ∆tis some in-

termediate time within the growth regime. The division

by (∆t)βremoves the systematic increase of the ﬂuctua-

tions in time so that, remarkably, the PDF of ϕireaches

a universal, time-independent form [7, 18, 19]. Important

examples in the kinetic roughening literature are e.g. the

Gaussian distribution for the linear EW equation [8, 9]

and a TW PDF (whose precise form depends, e.g., on

boundary conditions) for the KPZ equation [7, 18, 19].

Our main numerical ﬁndings can be summarized as

follows: 1) for odd (Kuramoto) coupling the scaling

Γ(∆φ) Odd Generic (non-odd)

Continuum Columnar EW [51] Columnar KPZ [54]

Scaling Super-rough [48] Faceted [53]

Exponents α= 3/2, z = 2 α≈1.07, z ≈1.39

Anom. Exp. αs= 3/2, αloc = 1 αs≈1.40, αloc ≈0.96

PDF Gaussian TW

Cφ(r, t) Larkin model Larkin model

TABLE I. Summary of correspondences between oscil-

lator models and kinetic roughening equations, and

main scaling and ﬂuctuation properties. Depending on

whether the coupling is odd or generic (non-odd) we ﬁnd dif-

ferent continuum approximations (corresponding to the two

main models of kinetic roughening with columnar noise), with

diﬀerent forms of anomalous scaling, exponent values, and

ﬂuctuation PDF. The covariances take the same form, how-

ever. References are given to works on the corresponding

interface equations. Results on the scaling exponents are con-

tained there and conﬁrmed by our simulations for the oscilla-

tor models. Results quoted for the PDF and covariance are

obtained from our simulations for the oscillator models.

is super-rough, the exponents and scaling Ansatz are

those of the EW equation with columnar noise (Larkin

model), and the ﬂuctuations follow a Gaussian PDF; 2)

for generic couplings the scaling is faceted, the exponents

and the scaling Asantz are those of the KPZ equation

with columnar noise, and the ﬂuctuations follow a TW

PDF. The covariance (11), however, is that of the Larkin

model in all cases. Thus, the case of non-odd coupling

seems to be the ﬁrst known example of kinetic roughen-

ing displaying TW statistics but not an Airy covariance.

These results are based on a 1D model of phase oscillators

described and explored in Sec. III, whose scaling and ﬂuc-

tuations are studied in Secs. IV (for odd coupling) and

V (for non-odd coupling).

These ﬁndings, together with some more speciﬁc de-

tails, are summarized in Table I. The exponents are di-

vided into two classes: the standard exponents αand z,

and the anomalous-scaling exponents αsand αloc. In the

case of odd coupling, all these exponents are simply read

from the exact Larkin model solution (see e.g. [51]). In

the case of generic couplings, they are obtained numeri-

cally, and what is presented is a rough approximation, as

they slightly change with diﬀerent coupling parameters.

Moreover, they may depend on other nonuniversal de-

tails according to what is known on the scaling behavior

of the KPZ equation with columnar noise [8, 54, 63, 64].

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NonequilibriumcriticalitydrivenbyKardar-Parisi-ZhanguctuationsinthesynchronizationofoscillatorlatticesRicardoGutierrez1andRodolfoCuerno11GrupoInterdisciplinardeSistemasComplejos(GISC),DepartamentodeMatematicas,UniversidadCarlosIIIdeMadrid,28911Leganes,Madrid,SpainThesynchronizationofoscillatorens...

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Nonequilibrium criticality driven by Kardar-Parisi-Zhang uctuations in the synchronization of oscillator lattices Ricardo Guti errez1and Rodolfo Cuerno1

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