Papers in Physics, vol. 2, art. 020008 (2010)

Received: 20 October 2010, Accepted: 1 December 2010
Edited by: A. Vindigni
Reviewed by: A. A. Fedorenko, CNRS-Lab. de Physique, ENS de Lyon, France.
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.020008

www.papersinphysics.org

ISSN 1852-4249

Anisotropic finite-size scaling of an elastic string at the depinning
threshold in a random-periodic medium

S. Bustingorry,1∗A. B. Kolton1†

We numerically study the geometry of a driven elastic string at its sample-dependent
depinning threshold in random-periodic media. We find that the anisotropic finite-size
scaling of the average square width w2 and of its associated probability distribution are
both controlled by the ratio k = M/Lζdep , where ζdep is the random-manifold depinning
roughness exponent, L is the longitudinal size of the string and M the transverse peri-
odicity of the random medium. The rescaled average square width w2/L2ζdep displays a
non-trivial single minimum for a finite value of k. We show that the initial decrease for
small k reflects the crossover at k ∼ 1 from the random-periodic to the random-manifold
roughness. The increase for very large k implies that the increasingly rare critical configu-
rations, accompanying the crossover to Gumbel critical-force statistics, display anomalous
roughness properties: a transverse-periodicity scaling in spite that w2 � M , and sublead-
ing corrections to the standard random-manifold longitudinal-size scaling. Our results are
relevant to understanding the dimensional crossover from interface to particle depinning.

I. Introduction

The study of the static and dynamic properties of
d-dimensional elastic interfaces in d+1-dimensional
random media is of interest in a wide range of phys-
ical systems. Some concrete experimental exam-
ples are magnetic [1–4] or ferroelectric [5,6] domain
walls, contact lines of liquids [7], fluid invasion in
porous media [8, 9], and fractures [10, 11]. In all
these systems, the basic physics is controlled by the
competition between quenched disorder (induced
by the presence of impurities in the host materials)
which promotes the wandering of the elastic object,
against the elastic forces which tend to make the
elastic object flat. One of the most dramatic and

∗E-mail: sbusting@cab.cnea.gov.ar
†E-mail: koltona@cab.cnea.gov.ar

1 CONICET, Centro Atómico Bariloche, 8400 San Carlos
de Bariloche, Ŕıo Negro, Argentina.

worth understanding manifestations of this compe-
tition is the response of these systems to an external
drive.

The mean square width or roughness of the inter-
face is one of the most basic quantities in the study
of pinned interfaces. In the absence of an external
drive, the ground state of the system is disordered
but well characterized by a self-affine rough geome-
try with a diverging typical width w ∼ Lζeq , where
L is the linear size of the elastic object and ζeq
is the equilibrium roughness exponent. When the
external force is increased from zero, the ground
state becomes unstable and the interface is locked
in metastable states. To overcome the barriers sep-
arating them and reach a finite steady-state veloc-
ity v it is necessary to exceed a finite critical force,
above which barriers disappear and no metastable
states exist. For directed d-dimensional elastic in-
terfaces with convex elastic energies in a D = d + 1
dimensional space with disorder, the critical point

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

is unique, characterized by the critical force F = Fc
and its associated critical configuration [12]. This
critical configuration is also rough and self-affine
such that w ∼ Lζdep with ζdep the depinning rough-
ness exponent. When approaching the threshold
from above, the steady-state average velocity van-
ishes like v ∼ (F −Fc)β and the correlation length
characterizing the cooperative avalanche-like mo-
tion diverges as ξ ∼ (F −Fc)−ν for F > Fc, where
β is the velocity exponent and ν is the depinning
correlation length exponent [13–16]. At finite tem-
perature and for F � Fc, the system presents an
ultra-slow steady-state creep motion with universal
features [17, 18] directly correlated with its multi-
affine geometry [19,20]. At very small temperatures
the absence of a divergent correlation length below
Fc shows that depinning must be regarded as a non-
standard phase transition [20, 21] while exactly at
F = Fc, the transition is smeared-out with the ve-
locity vanishing as v ∼ Tψ, with ψ, the so-called
thermal rounding exponent [22–27].

During the last years, numerical simulations have
played an important role to understand the physics
behind the depinning transition thanks to the de-
velopment of powerful exact algorithms. In partic-
ular, the development of an exact algorithm able to
target efficiently the critical configuration and crit-
ical force for a given sample [28, 29] has allowed to
study, precisely, the self-affine rough geometry at
depinning [7, 28–31], the sample-to-sample critical
force distribution [32], the critical exponents of the
depinning transition [26, 27, 33], the renormalized
disorder correlator [34], and the avalanche-size dis-
tribution in quasistatic motion [35]. Moreover, the
same algorithm has allowed to study, precisely, the
transient universal dynamics at depinning [36, 37],
and an extension of it has allowed to study low-
temperature creep dynamics [20, 21].

In practice, the algorithm for targeting the crit-
ical configuration [28, 29] has been numerically
applied to directed interfaces of linear size L dis-
placing in a disordered potential of transverse di-
mension M, applying periodic boundary conditions
in both directions in order to avoid border effects.
This is thus equivalent to an elastic string displac-
ing in a disordered cylinder. The aspect ratio be-
tween longitudinal L and transverse M periodic-
ities must be carefully chosen, in order to have
the desired thermodynamic limit corresponding to
a given experimental realization. In Ref. [32] it

was indeed shown that the critical force distribu-
tion P(Fc) displays three regimes associated with
M: (i) At very small M compared with the typical
width Lζdep of the interface, the interface wraps the
computational box several times in the transverse
direction, as shown schematically in Fig. 1(b), and
therefore the periodicity of the random medium is
relevant and P(Fc) is Gaussian; (ii) At very large
M compared with Lζdep , as shown schematically in
Fig. 1(c), periodicity effects are absent but then the
critical force, being the maximum among many in-
dependent sub-critical forces, obeys extreme value
statistics and P(Fc) becomes a Gumbel distribu-
tion; (iii) In the intermediate regime, where M ≈
Lζdep and periodicity effects are still irrelevant, as
shown schematically in Fig. 1(a), the distribution
function is in between the Gaussian and the Gum-
bel distribution. It has been argued that only the
last case, where M ≈ Lζdep , corresponds to the
random-manifold depinning universality class (pe-
riodicity effects absent) with a finite critical force
in the thermodynamic limit L,M → ∞. This cri-
terion does not give, however, the optimal value
of the proportionality factor between M and Lζdep ,
and must be modified at finite velocity since the
crossover to the random-periodic universality class
at large length-scales depends also on the veloc-
ity [38]. To avoid this problem, it has been there-
fore proposed to define the critical scaling in the
fixed center of mass ensemble [39]. The crossover
from the random-manifold to the random-periodic
universality class is, however, physically interest-
ing, as it can occur in periodic elastic systems such
as elastic chains. Remarkably, although the map-
ping from a periodic elastic system (with given lat-
tice parameter) in a random potential to a non-
periodic elastic system (such as an interface) in
a random potential with periodic boundary con-
ditions is not exact, it was recently shown that the
lattice parameter does play the role of M for elastic
interfaces with regard to the geometrical or rough-
ness properties [38]. Since the periodicity can often
be experimentally tuned in such periodic systems
it is thus worth studying in detail the geometry
of critical interfaces of size L as a function of M
with periodic boundary conditions, and thus com-
plement the study of the critical force in such sys-
tems [32].

In this paper, we study in detail, using numerical
simulations, the geometrical properties of the one-

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

Figure 1: (a) Elastic string driven by a force F in
a random-periodic medium with periodic boundary
conditions. It is described by a displacement field
u(z) and has a mean width w. The anisotropic
finite-size scaling of width fluctuations are con-
trolled by the aspect-ratio parameter k = M/Lζdep ,
with ζdep the random-manifold roughness exponent
at depinning. In the case k � 1 (b) periodicity ef-
fects are important, while when k � 1 (c) they
are not important but the roughness scaling of the
critical configuration is anomalous.

dimensional interface or elastic string critical con-
figuration in a random-periodic pinning potential
as a function of the aspect ratio parameter k, con-
veniently defined as k = M/Lζdep . We show that k
is indeed the only parameter controlling the finite-
size scaling (i.e. the dependence of observables with
the dimensions L and M) of the average square
width and its sample-to-sample probability distri-
bution. The scaled average square width w2L−2ζdep
is described by a universal function of k displaying
a non-trivial single minimum at a finite value of
k. We show that while for small k this reflects the
crossover at k ∼ 1 from the random-periodic to
the random-manifold depinning universality class,
for large k it implies that in the regime where the
depinning threshold is controlled by extreme value
(Gumbel) statistics, critical configurations also be-
come rougher, and display an anomalous roughness
scaling.

II. Method

The model we consider here is an elastic string in
(1+1) dimensions described by a single valued func-
tion u(z,t), which gives the transverse displace-
ment u as a function of the longitudinal direction z
and the time t [see Fig. 1(a)]. The zero-temperature
dynamics of the model is given by

γ ∂tu(z,t) = c∂
2
zu(z,t) + Fp(u,z) + F, (1)

where γ is the friction coefficient and c the elas-
tic constant. The first term in the right hand side
derives from an harmonic elastic energy. The ef-
fects of a random-bond type disorder is given by the
pinning force Fp(u,z) = −∂uU(u,z). The disorder
potential U(u,z) has zero average and sample-to-
sample fluctuations given by

[U(u,z) −U(u′,z′)]2 = δ(z −z′) R2(u−u′), (2)

where the overline indicates average over disorder
realizations and R(u) stands for a correlator of fi-
nite range rf [18]. Finally, F represents the uni-
form external drive acting on the string. Physi-
cally, this model can phenomenologically describe,
for instance, a magnetic domain wall in a thin film
ferromagnetic material with weak and randomly lo-
cated imperfections [1], being F proportional to an
applied external magnetic field pushing the wall in
the energetically favorable direction.

In order to numerically solve Eq. (1), the system
is discretized in the z-direction in L segments of
size δz = 1, i.e. z → j = 0, ...,L− 1, while keeping
uj(t) as a continuous variable. To model the con-
tinuous random potential, a cubic spline is used,
which passes through M regularly spaced uncor-
related Gaussian number points [30]. For the nu-
merical simulations performed here we have used,
without loss of generality, γ = 1, c = 1 and rf = 1
and a disorder intensity R(0) = 1. In both spatial
dimensions we have used periodic boundary condi-
tions, thus defining a L×M system.

The critical configuration uc(z) and force Fc are
defined from the pinned (zero-velocity) configura-
tion with the largest driving force F in the long
time limit dynamics. They are thus the real solu-
tions of

c∂2zu(z) + Fp(u,z) + F = 0, (3)

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

such that for F > Fc there are no further real so-
lutions (pinned configurations). Middleton theo-
rems [12] assure that for Eqs. (3) the solution ex-
ists and it is unique for both uc(z) and Fc, and that
above Fc the string trajectory in an L dimensional
phase-space is trapped into a periodic attractor (for
a system with periodic boundary conditions as the
one we consider). In other words, the critical con-
figuration is the marginal fixed point solution or
critical state of the dynamics, being Fc the criti-
cal point control parameter of a Hopf bifurcation.
Solving the L-dimensional system of Eqs. (3) for
large L directly is a formidable task, due to the
non-linearity of the pinning force Fp. On the other
hand, solving the long-time dynamics at different
driving forces F to localize Fc and uc is very in-
efficient due to the critical slowing down. Fortu-
nately, Middleton theorems, and in particular the
“non-passing rule”, can be used again to devise a
precise and very efficient algorithm which allows to
obtain the critical force Fc and the critical configu-
ration ucj for each independent disorder realization
iteratively without solving the actual dynamics nor
directly inverting the system of Eqs. (3) [30]. Once
the critical force and the critical configuration are
determined with this algorithm, we can compute
the different observables. In particular, the square
width or roughness of the string at the critical point
for a given disorder realization is defined as

w2 =
1
L

L−1∑
j=0

[
ucj −

1
L

L−1∑
k=0

uck

]2
. (4)

Computing w2 for different disorder realizations al-
lows us to compute its disorder average w2 and the
sample-to-sample probability distribution P(w2).
In addition, the average structure factor associated
to the critical configuration is

Sq =
1
L

∣∣∣∣∣∣
L−1∑
j=0

ucj e
−iqj

∣∣∣∣∣∣
2

, (5)

where q = 2πn/L, with n = 1, ...,L − 1. One
can show, using a simple dimensional analysis, that
given a roughness exponent ζ, such that w2 ∼ L2ζ,
the structure factor behaves as S(q) ∼ q−(1+2ζ)
for small q, thus yielding an estimate to ζ with-
out changing L. To compute averages over disor-
der and sample-to-sample fluctuations, we consider

Figure 2: The scaling of w2 for the critical con-
figuration at different M values as indicated. The
curves for M = 64 and 16384 are shifted upwards
for clarity. The dashed and dotted lines are guides
to the eye showing the expected slopes correspond-
ing to the different roughness exponents.

between 103 and 104 independent disorder realiza-
tions depending on the size of the system.

III. Results

i. Roughness at the critical point

Figure 2 shows the scaling of the square width of
the critical configuration w2 with the longitudinal
size of the system L for L = 32, 64, 128, 256, 512 and
different values of M. When M is small, M = 8, for
all the L values shown we observe w2 ∼ L2ζL with
ζL = 1.5, corresponding to the Larkin exponent in
(1 + 1) dimensions. This value is different from the
value ζdep = 1.25 [33, 40] expected for the random-
manifold universality class, and is thus indicating
that the periodicity effects are important for this
joint values of M and L. This situation is schemat-
ically represented in Fig. 1(b). This result is a
numerical confirmation of the two-loop functional
renormalization group result of Ref. [16] which
shows that the ζ = 0 fixed point, leading to a uni-
versal logarithmic growth of displacements at equi-
librium is unstable. The fluctuations are governed,
instead, by a coarse-grained generated random-
force as in the Larkin model, yielding a roughness
exponent ζL = (4−d)/2 in d dimensions [16], which
agrees with our result for d = 1. We can thus say

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

Figure 3: Structure factor of the critical configura-
tion for L = 256 and different M values, as indi-
cated. The curves for M = 64 and 16384 are shifted
upwards for clarity. The dashed and dotted lines
are guides to the eye showing the expected slopes
corresponding to the different roughness exponents.

that for small enough M (compared to L) the sys-
tem belongs to the same random-periodic depin-
ning universality class as charge density wave sys-
tems [14, 41], which strictly correspond to M = 1.

When M is large, on the other hand, M =
16384 in Fig. 2, for all the L values consid-
ered the exponent is consistent with ζdep, of the
random-manifold universality class. This situation
is schematically represented in Fig. 1(c), and we
will show later that, for this elongated samples, the
effects of extreme value statistics are already visi-
ble.

For intermediate values of M, such as M =
64 in Fig. 2, we can observe the crossover in
the scale-dependent roughness exponent ζ(L) ∼
1
2
d log w2

d log L
changing from ζdep to ζL as L increases,

as indicated by the dashed and dotted lines.
This crossover, from the random-manifold to the
random-periodic depinning geometry, occurs at a
characteristic distance l∗ ∼ M1/ζdep , when the
width in the random-manifold regime reaches the
transverse dimension or periodicity M. At finite
velocity, this crossover length remains constant up
to a non-trivial characteristic velocity and then de-
creases with increasing velocity [38].

The above mentioned geometrical crossover can
be studied in more details through the analysis of
the structure factor S(q), for a line of fixed size
L. In Fig. 3 we show S(q) for L = 256 and M =

Figure 4: Scaling of the structure factor of the crit-
ical configuration for L = 256 and different values
of the transverse size M = 2p with p = 3, 4, ..., 14
M. Although the values of the two exponents are
very close, the change in the slope of the scaling
function against the scaling variable x = q M1/ζdep
is clearly observed.

8, 64, 16384. For the intermediate value M = 64 a
crossover between the two regimes is visible, and
can be described by

Sq ∼
{
q−(1+2ζL) q � q∗,
q−(1+2ζdep) q � q∗.

(6)

with q∗ expected to scale as q∗ ∼ l∗−1 ∼ M−1/ζdep .
Therefore, the structure factor should scale as
SqM

−(2+1/ζdep) = H(x), where the scaled variable
is x = q M1/ζdep ∼ q/q∗ and the scaling function
behaves as

H(x) ∼
{
x−(1+2ζL) x � 1,
x−(1+2ζdep) x � 1.

(7)

The collapse of Fig. 4 for L = 256 and different val-
ues of M = 2p with p = 3, 4, ..., 14 shows that this
scaling form is a very good approximation. How-
ever, as we show below, small corrections can be
expected fully in the random-manifold regime in
the large ML−ζdep limit of very elongated samples.

In Fig. 5(a), we show w2 as a function of the
transverse periodicity M for different values of the
longitudinal periodicity L. Remarkably, w2 is a
non-monotonic function of M. For small M it de-
creases towards an L dependent minimum m∗, and
then increases with increasing M, in the regime
where the extreme value statistics starts to affect
the distribution of the critical force [32]. Since

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

Figure 5: (a) Squared width of the critical con-
figuration as a function of M for different system
sizes L as indicated. (b) Scaling of the width in
(a), showing that the relevant control parameter
is M/Lζdep . The dashed line in (a) and (b) corre-
sponds to w2 = M2, which is always to the left of
the minimum of w2 occurring at k∗ = m∗L−ζdep .
The solid line indicates k2(1−ζL/ζdep) which is the
behavior expected purely from the random-periodic
to random-manifold crossover at the characteristic
distance l∗ ∼ M1/ζdep .

the only typical transverse scale in Fig. 5(a) is
set by the minimum m∗, we can expect w2 ∼
m∗2G(M/m∗) with G(x) some universal function.
On the other hand, since the only relevant char-
acteristic length-scale of the problem is set by the
crossover between the random-periodic regime and
the random-manifold regime, we can simply write
m∗ ∼ Lζdep and therefore

w2 L−2ζdep ∼ G(M L−ζdep ). (8)

This scaling form is confirmed in Fig. 5(b)
and shows that the aspect-ratio parameter k =
ML−ζdep fully controls the anisotropic finite-size
scaling of the problem. It is worth, however, not-
ing some interesting consequences of the result of
Fig. 5(b), as we describe below.

Since at very small k the interface is in the
random-periodic regime, Eq. (8) should led to w2 ∼
L2ζL and therefore one deduces that,

G(k) ∼ k2(1−ζL/ζdep), k � k∗, (9)

where k∗ = m∗L−ζdep . The fact that the random-
periodic roughness exponent ζL = 3/2 is larger than
the random-manifold one ζdep ≈ 5/4 consequently
implies an initial decrease of G(k) as G(k) ∼ k−2/5,
as shown in Fig. 5(b) by the solid line. Periodicity
effects, or the crossover from random-periodic to
random-manifold, thus explain the initial decrease
of G(k) seen in Fig. 5(b), or the initial decrease of
w2 against M for fixed L, seen in Fig. 5(a). At
this respect, it is then worth noting that the line
w2 = M2, shown by a dashed line, lies completely
in the regime k < k∗ implying that the naive cri-
terion w2 < M2 is not enough to avoid period-
icity effects, and to have the system fully in the
random-manifold regime. As we show later, this
is related with the shape of the probability distri-
bution of P(w2) which displays sample-to-sample
fluctuations of the order of the average w2.

The presence of a minimum at k∗ in the function
G(k) and in particular its slower than power-law
increase for k > k∗ is non-trivial and constitutes
one of the main results of the present work. This
result shows that corrections to the standard scal-
ing w2 ∼ Lζdep may arise from the aspect-ratio de-
pendence of the prefactor G(k). On the one hand,
w2 now grows with M for L fixed, in spite that
w2 � M2, i.e. transverse-size/periodicity scaling is
present. On the other hand, the scaling of w2 with

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

L is slower in this regime, due to subleading scaling
corrections coming from G(k). The precise origin
of these interesting leading and subleading correc-
tions in the finite-size anisotropic scaling are highly
non-trivial. Since the critical configurations in this
regime have the constant roughness exponent ζdep
of the random-manifold universality class, the slow
increase of G(k) cannot be attributed to a geomet-
rical crossover effect, as for the case k < k∗. How-
ever, we might relate this effect to the crossover
in the critical force statistics, from Gaussian to
Gumbel, in the k � k∗ limit [32]. In the Gum-
bel regime, the average critical force is expected to
increase as Fc ∼ log(M/L

ζ
dep) ≡ log k [39], since

the sample critical force can be roughly regarded
as the maximum among M/Lζdep independent sub-
critical forces and configurations [32]. The increase
in the critical force might be therefore correlated
with the slow increase of roughness. The physi-
cal connection between the two is subtle though,
since a large critical force in a very elongated sam-
ple could be achieved both by profiting very rare
correlated pinning forces such as accidental colum-
nar defects, or by profiting very rare non-correlated
strong pinning forces. Since in the first case the
critical configuration would be more correlated and
in general less rough than for less elongated sam-
ples (smaller k), contrary to our numerical data of
Fig. 5(b), we think that the second cause is more
plausible. We can thus think that in the k � k∗
limit of extreme value statistics of Fc, the effective
disorder strength on the critical configuration in-
creases with k. This might be translated into the
universal function G(k), such that w2 ≈ L2ζdepG(k)
can increase for increasing values of k at fixed L in
such regime. A quantitative description of these
scaling corrections remains an open challenge.

ii. Distribution function

We now analyze sample-to-sample fluctuations of
the square width w2 by computing its probability
distribution P(w2). This property is relevant as
w2 fluctuates even in the thermodynamic limit for
critical interfaces with a positive roughness expo-
nent [42]. It has been computed for models with
dynamical disorder such as random-walk [43] or
Edwards–Wilkinson interfaces [44, 45], the Mullins
Herrings model [46] and for non-Markovian Gaus-
sian signals in general [47, 48]. It has also been

Figure 6: Scaling function Φ(x) for L = 256 and
different values of M = 8, 128, 2048, 16384, which
shows the change with the transverse size M.

calculated for non-linear models such as the one-
dimensional Kardar–Parisi–Zhang model [49, 50]
and for the quenched Edwards–Wilkinson model at
equilibrium [51].

In particular, the probability distribution P(w2)
of critical interfaces at the depinning transition
was studied analytically [52], numerically [31] and
also experimentally for contact lines in partial
wetting [7]. Remarkably, non-Gaussian effects in
depinning models are found to be smaller than
0.1% [31, 52], thus showing that P(w2) is strongly
determined by the self-affine (critical) geometry it-
self, rather than by the particular mechanism pro-
ducing it. As in all the above mentioned systems
the width distribution P(w2) at different universal-
ity classes of the depinning transition was found to
scale as

w2P(w2) ≈ Φζ
(
w2

w2

)
. (10)

with Φζ an universal function, which only depends
on the roughness exponent ζ and on boundary con-
ditions when the global width is considered [47,48].
In this way, w2 is the only characteristic length-
scale of the system, absorbing the system longitu-
dinal size L, and all the non-universal parameters
of the model such as the elastic constant of the
interface, the strength of the disorder and/or the
temperature. Since Φζ can be easily generated us-
ing non-Markovian Gaussian signals [53], the quan-
tity w2P(w2) is a good observable to extract the

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

Figure 7: Scaling function Φ(x) for different val-
ues of L = 32, 64, 128, 256 while keeping (a) k =
M/Lζdep ≈ 1 and (b) k = M/Lζdep ≈ 0.025.
The dotted line corresponds to the scaling function
of the non-disordered Edwards–Wilkinson equa-
tion [43], while the continuous and dashed lines cor-
respond to the scaling functions of Gaussian signals
with ζ = 1.25 and ζ = 1.5, respectively [31, 53].

roughness exponent of a critical interface from ex-
perimental data.

In Fig. 6, we show how the scaled distribution
function Φ(x) ≡ w2 P(x w2) looks like for the de-
pinning transition in a random-periodic medium for
a fixed value L = 256 and different values of M. We
see that Φ(x) depends on M for small M but con-
verges to a fixed shape for large M. We also note
that for all M Φ(x) extends appreciably beyond
x = 1 explaining why the criterion w2 . M2 is not
enough to be fully in the random-manifold regime,
as noted in Fig. 5.

In Fig. 7, we show the scaling function Φ(x) for
different values of L and M but fixing the aspect-
ratio parameter k = M/Lζdep , k ≈ 1 > k∗ in

Fig. 7(a) and k ≈ 0.025 � k∗ in Fig. 7(b), with
k∗ the minimum of w2. Since data for the same
k practically collapses into the same curve, we can
write for our case:

w2P(w2) = Φ
(
w2

w2
,k

)
. (11)

Therefore, the anisotropic scaling of the probability
distribution is fully controlled by k, as it was found
for w2.

In Figs. 7(a) and (b), we also show the uni-
versal functions ΦζL and Φζdep generated using
non-Markovian Gaussian signals [31, 53], and for
comparison we also show Φ1/2 corresponding to
the Markovian periodic Gaussian signal or the
Edwards–Wilkinson equation [43]. Comparing this
with the collapsed data for depinning, we see that
the function Φ

(
w2

w2
,k

)
respects the limits

Φ (x,k → 0) = ΦζL (x),
Φ (x,k & k∗) ≈ Φζdep (x), (12)

as expected from the existence of the geometric
crossover between the roughness exponents ζL for
k → 0 and ζdep for k > k∗. For intermediate values
k < k∗, however, Φ

(
w2

w2
,k

)
does not necessarily

coincide with the one of a Gaussian signal func-
tion Φζ for a given ζ, since the critical configura-
tion includes a crossover length l∗ . L. Whether
multi-affine or effective exponent self-affine non-
Markovian Gaussian signals can be used to describe
satisfactorily these intermediate cases is an inter-
esting open issue.

IV. Conclusions

We have numerically studied the anisotropic finite-
size scaling of the roughness of a driven elastic
string at its sample-dependent depinning thresh-
old in a random medium with periodic bound-
ary conditions in both the longitudinal and trans-
verse directions. The average square width w2 and
its probability distribution are both controlled by
the parameter k = M/Lζdep . A non-trivial sin-
gle minimum for a finite value of k was found in
w2/L2ζdep . For small k, the initial decrease of w2 re-
flects the crossover from the random-periodic to the
random-manifold roughness. For very large k, the
growth with k implies that the crossover to Gumbel

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Papers in Physics, vol. 2, art. 020008 (2010) / S. Bustingorry et al.

statistics in the critical forces induces corrections
to G(k), that grow with k, to the string rough-
ness scaling w2 ≈ G(k)L2ζdep . These increasingly
rare critical configurations thus have an anoma-
lous roughness scaling: they have a transverse-
size/periodicity scaling in spite that its width is
w2 � M2, and subleading (negative) corrections
to the standard random-manifold longitudinal-size
scaling.

Our results could be useful for understanding
roughness fluctuations and scaling in finite ex-
perimental systems. The crossover from random-
periodic to random-manifold roughness could be
studied in periodic elastic systems with variable
periodicity, such as confined vortex rows [54] and
single-files of macroscopically charged particles [55]
or colloids [56], with additional quenched disor-
der. The rare-event dominated scaling corrections
to the interface roughness scaling could be studied
in systems with a large transverse dimension, such
as domain walls in ferromagnetic nanowires [57].
For the later case, it would be interesting to have
a quantitative theory, making the connection be-
tween the extreme value statistics of the depinning
threshold and the anomalous scaling corrections to
the roughness of such rare critical configurations.
This would allow to understand the dimensional
crossover, from interface to particle depinning.

Acknowledgements - This work was sup-
ported by CNEA, CONICET under Grant No.
PIP11220090100051, and ANPCYT under Grant
No. PICT2007886. A. B. K. acknowledges Uni-
versidad de Barcelona, Ministerio de Ciencia e In-
novación (Spain) and Generalitat de Catalunya for
partial funding through I3 program.

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