Julien Charreau

Yan Chen

Stuart Gilder

Laurie Barier

S Dominguez

J.-P Avouac

S Sen

M Jolivet

Y Li

W Wang

to Neogene continental sediments from two sections located in the Kuche basin at the northern edge of the Tarim basin. To support their conclusion they reinterpreted a magnetostratigraphic

Keywords: Magnetostratigraphy, Tianshan, sedimentation rates, AMS, uplift

discusses the Cenozoic uplift history of the Tianshan Mountains by studying the magnetostratigraphy of Paleogene

Understanding how and when the Tianshan mountains were built not only improves our knowledge of mountain building processes in general but also how deformation has propagated to form the vast area under the influence of the India-Asia collision. Numerous studies over the last few decades have been dedicated to unraveling the Tertiary history of this impressive mountain range (e.g., [START_REF] Avouac | Active Thrusting and folding along the northern Tien Shan and late Cenozoic rotation of the Tarim relative to Dzungaria and Kazakhstan[END_REF], [Hendrix et al., 1994], [Métivier and Gaudemer, 1997], [Sobel and Dumitru, 1997], [START_REF] Burchfiel | Crustal Shortening on the Margins of the Tian Shan, Xinjiang, China[END_REF], [START_REF] Bullen | Late Cenozoic tectonic evolution of the northwestern Tien Shan: New age estimates for the initiation of mountain building[END_REF], [START_REF] Bullen | Building the Northern Tien Shan: Integrated Thermal, Structural, and Topographic Constraints[END_REF], [START_REF] Dumitru | Uplift, exhumation, and deformation in the Chinese Tian Shan[END_REF], [Sun et al., 2004], [Charreau et al., 2005], [START_REF] Charreau | Magnetostratigraphy of the Yaha section, Tarim Basin (China): 11 Ma acceleration in erosion and uplift of the Tianshan Mountains[END_REF] and [Sobel et al., 2006]). One commonly used tool is magnetostratigraphy, which can decipher the history of the Tianshan from changes in the deposition rates of the sediments shed from the mountains and deposited in the foreland Junggar and Tarim basins ( [START_REF] Bullen | Late Cenozoic tectonic evolution of the northwestern Tien Shan: New age estimates for the initiation of mountain building[END_REF], [START_REF] Bullen | Building the Northern Tien Shan: Integrated Thermal, Structural, and Topographic Constraints[END_REF], [Sun et al., 2004], [Charreau et al., 2005], [START_REF] Charreau | Magnetostratigraphy of the Yaha section, Tarim Basin (China): 11 Ma acceleration in erosion and uplift of the Tianshan Mountains[END_REF] and [START_REF]Magnetochronology of the Upper Cenozoic strata in the Southerstern Chinsese Tian Shan: rates of Pleistocene folding and thrusting[END_REF]).

Recently, Huang et al. (2006) published a magnetostratigraphic and rock magnetic study of Paleogene to Neogene continental sediments from two sections, separated by 2 km, located in the Kuche basin at the northern edge of the Tarim basin. These sections lie ~ 10 km north of the Yaha magnetostratigraphic section that we published in January of 2006 [START_REF] Charreau | Magnetostratigraphy of the Yaha section, Tarim Basin (China): 11 Ma acceleration in erosion and uplift of the Tianshan Mountains[END_REF]. Changes in sedimentation rates derived from the magnetostratigraphic dating and changes in anisotropy of magnetic susceptibility (AMS) parameters led Huang et al. (2006) to suggest that the southern Tianshan underwent uplift pulses at ~ 20 Ma, at ~ 16-17 Ma, and then a more regional phase at ~ 7 Ma. The timing of these uplift pulses differ from the one we identified at Yaha around 10 to 11 Ma. Moreover, Huang et al. (2006) reinterpreted our Yaha magnetostratigraphic column, proposing that it should be shifted 3 million years younger in time than we originally found. Our motivation to write this comment is because (1) we think the interpretations of the sedimentation rate changes proposed by Huang et al. (2006) for their sections are partially invalid, (2) we disagree with their reinterpretation of the age of the Yaha section, and (3) we think that the way Huang et al. (2006) interpret their AMS data is incorrect.

(1) Changes in sedimentation rates 2006) data for their section A, which overlaps the Yaha section in time. According to the authors, sedimentation rate was constant from 16 to 7 Ma, then increased at ca. 7 Ma. Their interpretation is shown as a thick dashed line in Fig. 1. On the other hand, one can fit two linear segments to their data with a break at 10 to 11 Ma, compatible with that identified at the nearby Yaha section. This new interpretation shows a relatively better fit when compared to their previous one. Whether or not a change in sedimentation rate occurred at 7 Ma is debatable because only four reversals define the break in slope, thus making the interpretation more tenuous. (2)Correlation of the Yaha magnetostratigraphic column We dated the Yaha magnetostratigraphic section between 12.6 Ma and 5.2 Ma. Our correlation suggests that sedimentation rate doubled at ~ 11 Ma. Also at ca. 11 Ma, an abrupt change in the AMS shape parameter T was observed. We interpreted these changes as the mark of increased uplift and concomitant erosion at that time. Thermochronological studies in the Tianshan also identify important changes at ca. 11 Ma ( [START_REF] Bullen | Late Cenozoic tectonic evolution of the northwestern Tien Shan: New age estimates for the initiation of mountain building[END_REF] and [START_REF] Bullen | Building the Northern Tien Shan: Integrated Thermal, Structural, and Topographic Constraints[END_REF]). Huang et al. (2006) proposed "an equally valid" correlation of our magnetostratigraphic scale with the reference geomagnetic polarity time scale (GPTS) [START_REF] Berggren | A revised geochronology and chronostratigraphy, Geochronology Time Sclaes and Global Stratigraphic Correlation[END_REF] between ~ 2 Ma (C2r.1n) and ~ 10 Ma (C5n.1n), which would make the Yaha section 3 million years younger in time than we proposed. Admittedly, in the absence of strong paleontological or radiochronological constraints, magnetostratgraphic correlation is a subjective, eye-based exercise based on pattern matching. Nevertheless, lithology, variations of sedimentation rate, statistical tests and basic diagrams can help quantify the quality of a given correlation. Huang et al. (2006) never presented depth/time diagrams or statistical analyses to argue why their reinterpretation should be considered better than that of [START_REF] Charreau | Magnetostratigraphy of the Yaha section, Tarim Basin (China): 11 Ma acceleration in erosion and uplift of the Tianshan Mountains[END_REF]. Fig. 2 shows the correlations of our data with the GPTS [START_REF] Berggren | A revised geochronology and chronostratigraphy, Geochronology Time Sclaes and Global Stratigraphic Correlation[END_REF] made by us (left) and Huang (right). Our correlation contains 2 chrons that do not exist in the GPTS and 1 missing chron. We note that the one missing chron comes from lower in the section where the sedimentation rates are lower (~ 20 cm/ka). Besides these three, we successfully matched each chron with the GPTS. On the other hand, the Huang et al. ( 2006) correlation contains 2 chrons that do not correlate with the GPTS and 4 missing chrons. Considering the extremely high sedimentation rates imposed by the Huang et al. ( 2006) correlation, exceeding 50 cm/ka (Fig. 2b), the number of missed chrons is suspect, representing more than 10% of 36 chrons identified in the Yaha section. An important outcome of the age of the sampled Yaha section directly bears on the sedimentation rates of the strata that lie above the sampled part of the section, which is dominated by the conglomerate-rich Xiyu Formation. According to Hubert-Ferrari et al. (2007) who analyzed seismic lines across the Yaha section, there are ~ 2000 meters from the top of the magnetostratigraphic section to the top of the sedimentary pile at Yaha. Our correlation dates the top of the magnetostratigraphic section at 5.2 Ma. This implies the average sedimentation rate from the top of the section to the top of the Xiyu Formation is ~ 38 cm/ka, which is very similar to the mean instantaneous sedimentation rate of the upper half of the Yaha section based on our correlation (43 cm/ka). Huang et al.'s reinterpretation is that the top of our magnetostratigraphic section is ca. 2 Ma. This means that the average sedimentation rate must be 100 cm/ka, which well exceeds published average sedimentation rates for the entire Asian theatre by a factor of two. This is highly improbable.

In sum, we reject the reinterpretation of the Yaha magnetostratigrahic correlation offered by Huang et al. (2006) because: (1) their correlation has two times the number of missed or unidentified chrons than ours (2) their correlation imposes more dramatic sedimentation rate variations, (3) their correlation predicts sedimentation rates about two-times higher than any previous study on continental sediments in Asia, and (4) extrapolation to the top of the sedimentary pile at Yaha imposes even more unrealistic sedimentation rates.

(3) Interpretation of Anisotropy of Magnetic Susceptibility Parameters

The magnetic fabrics of sediments yield information related to the deposition conditions acting when the sediments were deposited. Tectonic stress can overprint the sedimentary fabric, and thus magnetic fabrics of sediments can also be useful indicators of strain ([Pares et al., 1999], [Parès and Van der Pluijm, 2002], [Hrouda, 1991] and [Kanamatsu et al., 1996]). That the AMS data in the Huang et al. ( 2006) sections, as reflected by the principal axis directions, record strain is undisputable. However, Huang et al. (2006) interpret the age of the stress to be coeval with deposition; and thus, that the time-transgressive changes in the AMS parameters reflect the stress imposed on the sediments at the time of deposition. We object to this interpretation because there is no evidence to suggest that the sediments were progressively deformed. Detailed structural geology work performed at the nearby Yakeng anticline, which is a pure-shear detachment fold, has a well-constrained history of growth beginning at 5.5 Ma (Hubert-Ferrari et al., 2007). The fact that the section lying closer to the deformation front yields greater degrees of anisotropy and better defined fabrics is typical of that found in orogenic fronts (Pares et al., 1999). The sediments toward the bottom of the section are also more clay rich, which react differently to stress than the coarser-grained sediments closer to the top of the sequence. The same was found at the Subei section (Gilder et al., 2001).

Of the two sections (called A and B) sampled by Huang et al. (2006), Section B is characterized by declinations generally > 0° and lies closer to the Tianshan Mountains than Section A, which is characterized by declinations generally < 0°. Huang et al. (2006) argued that the change in declination within and between the two sections was progressive, with inferred clockwise rotation taking place from ca. 26 to 12 Ma, followed by counterclockwise rotation from 12 Ma to present. However, as pointed out by Gilder et al. (in press), an anticline is situated at the northern part of Section A, in-between the two sections (see Fig. 1 of Huang et al., 2006). Because anticlines are often an expression of deep-seated thrust faults (Hubert-Ferrari et al., 2007), this structural discontinuity makes it plausible that the relative rotation between the two sections is due to a differential vertical axis block rotation, younger than ca. 5.5 Ma, again consistent with the deformation history at the Yakeng anticline beginning at 5.5 Ma (Hubert-Ferrari et al., 2007). Moreover, paleomagnetic declinations from the Yaha section are identical to those from Section A, yet display no hint of a progressive rotation [START_REF] Charreau | Magnetostratigraphy of the Yaha section, Tarim Basin (China): 11 Ma acceleration in erosion and uplift of the Tianshan Mountains[END_REF]. We also note that there is a marked difference in AMS signature between the two sections that Huang et al. ( 2006) sampled. Thus, the abrupt changes in the AMS T, q and Pj parameters, which coincide with a structural break between the sections, as marked by an anticline, opens the question whether the differences are linked to the way stress was transmitted to the rocks at the different sections. Thus, we interpret the AMS fabric as reflecting a tectonic overprint, which is Pliocene or younger.

Conclusions

We do not agree with the way Huang et al. (2006) defined changes in the sedimentation rate in the upper part of their sections, and we think that our re-analysis of their data support a change at 10-11 Ma, compatible with that seen at the Yaha section. Their suggestion of an acceleration in sedimentation rates at ~ 7 Ma warrants more careful consideration. Changes in the AMS data do not reflect tectonic strain at the time of deposition as the authors believe, but instead are likely due to Pliocene-Pleistocene deformation. We dispute the reinterpretation by Huang et al. (2006) of the age of the Yaha section. Rather, from both studies we find that the Kuche Basin sediments recorded three main events at 20 Ma, 15 Ma and 10-11 Ma, which coincides with the conclusions of Charreau (2005).

Fig. 1

 1 Fig. 1 plots thickness versus time for the Huang et al. (2006) data for their section A, which overlaps the Yaha section in time. According to the authors, sedimentation rate was constant from 16 to 7 Ma, then increased at ca. 7 Ma. Their interpretation is shown as a thick dashed line in Fig.1. On the other hand, one can fit two linear segments to their data with a break at 10 to 11 Ma, compatible with that identified at the nearby Yaha section. This new interpretation shows a relatively better fit when compared to their previous one. Whether or not a change in sedimentation rate occurred at 7 Ma is debatable because only four reversals define the break in slope, thus making the interpretation more tenuous.

Fig. 1 .

 1 Fig. 1. Age vs. depth plot of section A of Huang et al. (2006) according to their correlation. The thick dashed line shows the best-fit according to Huang et al. that yields an acceleration in the sedimentation rates at ~ 7 Ma. The black line represents our preferred fit, which suggests an acceleration in the sedimentation rate at ~ 10-11 Ma.

Fig. 2 .

 2 Fig. 2. (a) Synthetic sedimentological column of the Yaha section (b) Age vs. depth plot of the Yaha magnetostratigraphic column section showing the correlations of Charreau et al. (2006) (black circles) and Huang et al. (2006) (gray circles). White boxes list the correlation coefficient (R 2 ) for the different segments (c) plot of sedimentation rate against time calculated based on the Charreau et al. (2006) correlation (left) and the Huang et al. (2006) correlation (right).

Fig. 2 .

 2 Fig. 2. (a) Synthetic sedimentological column of the Yaha section (b) Age vs. depth plot of the Yaha magnetostratigraphic column section showing the correlations of Charreau et al. (2006) (black circles) and Huang et al. (2006) (gray circles). White boxes list the correlation coefficient (R 2 ) for the different segments (c) plot of sedimentation rate against time calculated based on the Charreau et al. (2006) correlation (left) and the Huang et al. (2006) correlation (right).

Acknowledgements

This study was financed by the French INSU (Institut National des Sciences de l'Univers) program ECLISPE (Environnement et CLImat du Passé: hiStoire et Evolution), the Agence National pour la Recherche (ANR) program NT05-3-42850, the Franco-Chinese Program de Recherche Avancé (PRA) T05-02 and the Chinese project 2005CB422101.

Asia: From continental assembly to intracontinental deformation, Geological Society of America Memoir vol. 194, Boulder, Colorado (2001), pp. 71-99. Gilder et al., 2001 S. Gilder, Y. Chen and S. Sevket, Oligo-Miocene magnetostratigraphy and rock magnetism of the Xishuigou section, Subei (Gansu Province, western China) and implications for shallow inclinations in central Asia, J. Geophys. Res. 106 (12) (2001), pp. 30505-30521 Gilder et al., 2008 S.A. Gilder, J. Gomez, Y. Chen and J.P. Cogné, A New Paleogeographic Configuration of the Eurasian Landmass Resolves a Paleomagnetic Paradox of the Tarim Basin (China), Tectonics 27 (2008), p. TC1012. Hendrix et al., 1994 M.S. Hendrix, T.A. Dumitru and A.S. Graham, Late Oligocene-early Miocene unroofing in the Chinese Tian Shan: An early effect of the India-Asia collision, Geology 22 (1994), pp. 487-490 Hrouda, 1991 F. Hrouda, Models of magnetic variations in sedimentary thrust sheets, Tectonophysics 185 (1991), pp. 203-210. Huang et al., 2006 B. Huang, J.D.A. Piper, S. Peng, T. Liu, Z. Li, Q. Wang and R. Zhu, Magnetostratigraphic study of the Kuche Depression, Tarim Basin, and Cenozoic uplift of the Tian Shan Range, Western China, Earth Planet. Sci. Lett. 251 (3-4) (2006), pp. 346-364. Hubert-Ferrari et al., 2007 A. Hubert-Ferrari, J. Suppe, R. Gonzalez-Mieres and X. Wang, Mechanisms of active folding of the landscape (southern Tian Shan, China), J. Geophys. Res. 112, B03S09 (2007). Kanamatsu et al., 1996 T. Kanamatsu, E. Herroro-Bervera, A. Taira, S. Saito and J. Ashi, Magnetic fabric development in tertiary accretionary complex in the Boso and Miura Peninsulas of central Japan, Geophys. Res. Lett. 23 (5) (1996), pp. 471-474. Métivier and Gaudemer, 1997 F. Métivier and Y. Gaudemer, Mass transfer between eastern Tien Shan and adjacent basins (central Asia): constraints on regional tectonics, Geophys. J. Int. 128 (1997), pp. 1-17. Parès and Van der Pluijm, 2002 J.M. Parès and B. Van der Pluijm, Evaluating magnetic lineations (AMS) in deformed rocks, Tectonophysics 350 (2002), pp. 283-298. Pares et al., 1999 J.M. Pares, B.A. van der Pluijm and J. Dinares-Turell, Evolution of magnetic fabrics during incipient deformation of mudrocks (Pyrenees, northern Spain), Tectonophysics Applications of Magnetic Anisotropies to Fabric Studies of Rocks and Sediments vol. 307(1-2) (1999), pp. 1-14 Sobel and Dumitru, 1997 E.R. Sobel and T.A. Dumitru, Thrusting and exhumation around the margins of the western Tarim basin during India-Asia collision, J. Geophys. Res. 102 (B3) (1997), pp. 5043-5063. Sobel et al., 2006 E.R. Sobel, M. Oskin, D. Burbank and A.V. Mikolaichuk, Exhumation of basement-cored uplifts: Example of the Kyrgyz Range quantified with apatite fission track thermochronology, Tectonics 25 (2006). Sun et al., 2004 J. Sun, R. Zhu and J. Bowler, Timing of the Tianshan Mountains uplift constrained by magnetostratigraphic analysis of molasse deposits, Earth Planet. Sci. Lett. 219 (2004), pp. 239-253.


Cathelijne Ter Burg

Kay Jörg Wiese

Mean-Field Theories for Depinning and their Experimental Signatures

Mean-field theory is an approximation replacing an extended system by a few variables. For depinning of elastic manifolds, these are the position of its center of mass u, and the statistics of the forces F (u). There are two proposals how to model the latter: as a random walk (ABBM model), or as uncorrelated forces at integer u (discretized particle model, DPM). While for many experiments the ABBM model (in the literature misleadingly equated with mean-field theory) makes quantitatively correct predictions for the distributions of velocities, or avalanche size and duration, the microscopic disorder force-force correlations cannot grow linearly, and thus unboundedly as a random walk, with distance. Even the effective (renormalized) disorder forces which do so at small distances are bounded at large distances. We propose to model forces as an Ornstein Uhlenbeck process. The latter has the statistics of a random walk at small scales, and is uncorrelated at large scales. By connecting to results in both limits, we solve the model largely analytically, allowing us to describe in all regimes the distributions of velocity, avalanche size and duration. To establish experimental signatures of this transition, we study the response function, and the correlation function of position u, velocity u and forces F under slow driving with velocity v > 0. While at v = 0 force or position correlations have a cusp at the origin and then decay at least exponentially fast to zero, this cusp is rounded at a finite driving velocity. We give a detailed analytic analysis for this rounding by velocity, which allows us, given experimental data, to extract the timescale of the response function, and to reconstruct the force-force correlator at v = 0. The latter is the central object of the field theory, and as such contains detailed information about the universality class in question. We test our predictions by careful numerical simulations extending over up to ten orders in magnitude.

I. INTRODUCTION A. Mean-field theories

The framework of disordered elastic manifolds covers such diverse systems as contact-line depinning [START_REF] Le Doussal | Height fluctuations of a contact line: A direct measurement of the renormalized disorder correlator[END_REF], charge-density waves, magnetic domain walls [START_REF] Urbach | Interface depinning, self-organized criticality, and the Barkhausen effect[END_REF][START_REF] Durin | Scaling exponents for Barkhausen avalanches in polycrystalline and amorphous ferromagnets[END_REF][START_REF] Kim | Direct observation of Barkhausen avalanche in Co thin films[END_REF][START_REF] Durin | The Barkhausen effect[END_REF][START_REF] Lemerle | Domain wall creep in an Ising ultrathin magnetic film[END_REF], earthquakes [START_REF] Jagla | The mechanisms of spatial and temporal earthquake clustering[END_REF][START_REF] Kagan | Seismic moment distribution revisited: I. Statistical results[END_REF][START_REF] Fisher | Collective transport in random media: From superconductors to earthquakes[END_REF][START_REF] Fisher | Statistics of Earthquakes in Simple Models of Heterogeneous Faults[END_REF][START_REF] Paczuski | Universality in sandpiles, interface depinning, and earthquake models[END_REF][START_REF] Ben-Zion | Earthquake failure sequences along a cellular fault zone in a three-dimensional elastic solid containing asperity and nonasperity regions[END_REF][START_REF] Crisanti | Strongly intermittent chaos and scaling in an earthquake model[END_REF][START_REF] Burridge | Model and theoretical seismicity[END_REF][START_REF] Gutenberg | Earthquake magnitude, intensity, energy, and acceleration[END_REF][START_REF] Gutenberg | Frequency of earthquakes in California[END_REF], shear of micro-pillars [START_REF] Csikor | Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale[END_REF] and stretching of a knit [START_REF] Poincloux | Crackling dynamics in the mechanical response of knitted fabrics[END_REF]. Many of these experiments, or at least aspects thereof, are successfully described by mean-field theory. But what exactly is meant by mean-field theory? Let us define mean-field theory as a theory which reduces an extended system to a a single or few degrees of freedom. For depinning these are u, the center of mass of the interface, and the correlations of F (u), the forces acting on it. The center of mass of the interface follows the equation of motion

∂ t u(t) = m 2 [w -u(t)] + F u(t) . (1)
The first term is the force exerted by a confining well, equivalent to a Hookean spring with spring constant m 2 . The second term F (u) is a random force, possibly the derivative of a random potential, F (u) = -V (u). Specifying the correlations of F (u) specifies the system, and selects one mean-field theory. However, when the reader encounters the term "meanfield theory" in the literature, it is quite generally employed for a model where the forces perform a random walk,

∂ u F u = ξ(u), (2)
ξ(u)ξ(u ) = 2δ(u -u ).

(

) 3
This model was introduced in 1990 by Alessandro, Beatrice, Bertotti and Montorsi (ABBM) [START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. I. Theory[END_REF][START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. II. Experiments[END_REF] to describe magnetic domain walls. There F (u) are the "coercive magnetic fields" pinning the domain wall, which were observed experimentally to change with a seemingly uncorrelated function ξ(u) [START_REF] Vergne | Quelques aspects statistiques des processus d'aimantation dans les corps ferromagnétiques. Cas du déplacement d'une seule paroi de Bloch à 180 • dans un milieu monocristallin aléatoirement perturbé[END_REF].

The decision of ABBM [START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. I. Theory[END_REF] to model ξ(u) in Eq. ( 3) as a white noise is a strong assumption, a posteriori justified by the applicability to experiments [START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. II. Experiments[END_REF]. It means that F (u) has the statistics of a random walk.

Field theory [START_REF] Le Doussal | Functional renormalization group and the field theory of disordered elastic systems[END_REF][START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF][START_REF] Chauve | Renormalization of pinned elastic systems: How does it work beyond one loop?[END_REF] gives a more differentiated view: First of all, mean-field theory should be applicable for d = d c [START_REF] Fedorenko | Depinning transition at the upper critical dimension[END_REF][START_REF] Doussal | Higher correlations, universal distributions and finite size scaling in the field theory of depinning[END_REF], a case which contains magnets with strong dipolar interactions [START_REF] Durin | The Barkhausen effect[END_REF], earthquakes [START_REF] Fisher | Collective transport in random media: From superconductors to earthquakes[END_REF], and micro-pillar shear experiments [START_REF] Csikor | Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale[END_REF]. As F (u) has the statistics of a random walk, the (microscopic) force-force correlator of Eqs. ( 2)-( 3) is

∆ RW 0 (0) -∆ RW 0 (u -u ) := 1 2 [F (u) -F (u )] 2 = |u -u |.
(4) A linearly increasing correlation function is at the microscopic level predicted for the correlations of the potential, R(0) -R(u) := 1 2 [V (u)-V (0)] 2 , if the disorder is of the random-field type, the strongest microscopic disorder at our disposal [START_REF] Wiese | Functional renormalization for disordered systems: Basic recipes and gourmet dishes[END_REF][START_REF] Wiese | A unified theory for disordered elastic manifolds, depinning, avalanches, and sandpiles[END_REF]. We know of no microscopic mechanism to generate the correlations of Eq. ( 4). On the other hand, the effective (renormalized) force-force correlator ∆(u) has a cusp [START_REF] Wiese | A unified theory for disordered elastic manifolds, depinning, avalanches, and sandpiles[END_REF][START_REF] Middleton | Measuring functional renormalization group fixed-point functions for pinned manifolds[END_REF][START_REF] Rosso | Numerical calculation of the functional renormalization group fixed-point functions at the depinning transition[END_REF], so Eq. ( 4) with |∆ (0 + )| = 1 is an approximation, valid for small u. The ABBM model defined by Eqs. ( 1)-(3) should then be viewed as an effective theory, arriving after renormalization, and valid for small u only.

If indeed the microscopic disorder has the statistics of a random walk, then the force-force correlator (4) does not change under renormalization, as is easily checked by inserting it into the 1-loop or 2-loop flow equations for depinning [START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF][START_REF] Chauve | Renormalization of pinned elastic systems: How does it work beyond one loop?[END_REF]. Counting of derivatives for higher-order corrections proves that this statement persists to all orders in perturbation theory. Thus even an extended (non-MF) system where each degree of freedom sees a force which has the statistics of a random walk, the Brownian-force model (BFM) introduced in Ref. [START_REF] Doussal | Avalanche dynamics of elastic interfaces[END_REF], is

m 2 w j' F(u) j w' u FIG.
1. Geometric solution to find uw as a function of w for the DPM model of [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF].

stable under renormalization, and has a a roughness exponent ζ BFM = 4 -d, where d is the dimension of the elastic object. This was indirectly verified numerically in Ref. [START_REF] Zhu | The spatial shape of avalanches[END_REF].

Our discussion shows that the ABBM model ( 2)-( 3) is adequate only at small distances, but fails at larger ones, where the force-force correlator decorrelates. We therefore expect that at large distances it crosses over to a model of uncorrelated random forces. Such a model, which we term the discretized particle model (DPM), was introduced in Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF]. In this model, the random forces are modeled by needles at integer position, blocking the advance of the particle. Figure 1 shows this model, and how to geometrically construct the solution of Eq. ( 1): Draw a straight line f (u) := m 2 (u -w) of slope m 2 , intersecting the horizontal axis (F = 0) at u = w. As long as f (u) < F (u), the particles advances. Thus the motion is stopped at the first obstacle this line encounters. On Fig. 1, this is u(w) = j and u(w ) = j . Remains to specify the correlations of the random forces F (j), j ∈ N. As field theory in general supposes Gaussian disorder, we choose the F (j) to be Gaussian, with variance

F (j)F (j ) = δ j,j . (5)
In this article, we wish to study the crossover from ABBMdisorder given by Eqs. ( 2)-( 4), to the correlations [START_REF] Durin | The Barkhausen effect[END_REF]. We propose to model this by replacing Eq. ( 2) by an Ornstein-Uhlenbeck process,

∂ u F u = -F (u) + ξ(u) , (6)
keeping the noise correlations of Eq. ( 3). This leads to F (u) = 0 and microscopic correlations (see appendix A)

∆ OU 0 (u -u ) := F (u)F (u ) c = e -|u-u | . (7)
Note that the small-distance behavior of ∆ OU 0 (u -u ) is as in Eq. ( 4). On the other hand, at large scales, forces of the Ornstein-Uhlenbeck process are Gaussian and uncorrelated, as those of Eq. (5). To compare the two models, let us consider their microscopic disorder force-force correlator. For the DPM, with forces constant between integers 1 , it reads 1 To render F (u)F (u ) function of u -u only, the needle forces F (j), j ∈ N of Fig. 1 are extended to F (u) = F (j), j ≤ u < j + 1. The solutions uw of Fig. 1 change, uw → uw -δ, with 0 ≤ δ < 1, negligible for small m 2 .

∆ DPM 0 (u -u ) = F (u)F (u ) = max(1 -|u -u |, 0) . (8)
Both models have a linear (microscopic) cusp, with

-∆ 0 (0 + ) = 1 . (9)
On the other hand, the integral over their force-force correlations is different,

∞ -∞ du ∆ DPM 0 (u) = 1, (10)
∞ -∞ du ∆ OU 0 (u) = 2. ( 11
)
Our working hypothesis to be checked below is that the two models have the same universal large-scale properties, with the same scale, and without any additional parameter. Nevertheless, we expect that non-universal quantities such as the critical force might be shifted. Eq. ( 6) also serves as an effective theory for the crossover observed in systems of linear size L, from a regime with mL 1 described by an extended elastic manifold, to a single-particle regime described by the DPM. This crossover has indeed be seen in numerical simulations for a line with periodic disorder [START_REF] Kolton | Uniqueness of the thermodynamic limit for driven disordered elastic interfaces[END_REF].

B. The effective disorder and measurements

Force correlator at finite driving velocity

The question we are now turning to is the following: What can experiments teach us about the underlying field theory? Can the crossover be seen in an experiment?

Suppose the system is driven quasi-statically, i.e.

w = vt, (12)
and we wish to study the limit of v → 0. To do so, parameterize the solution of Eq. (1) as

u w := u(t). (13)
In this limit most of the time ∂ t u(x, t) = 0, and Eq. (1) yields

F u w = -m 2 (w -u w ). (14)
The geometric construction to find this solution is shown on Fig. 1. The critical force is defined as

f c (m) := -F (u w ) ≡ m 2 [w -u w ]. (15)
The signs are such that applying the external force f c (counted positive in the driving direction) overcomes the pinning forces F (u w ). At a finite driving velocity v, averaging Eq. ( 1) yields an additional term u = v, leading, at least for small velocity v, to

f c (m) := -F (u w ) ≡ m 2 [w -u w ] -v. ( 16
)
The effective disorder force-force correlator is defined as

∆(w -w ) := lim v→0 F (u w )F (u w ) c = lim v→0 m 4 [w -u w ][w -u w ] c . (17)
Note that if we consider an extended system, and u w is the center of mass position u w := 1 L d x u w (x), then there is an additional factor of L d on the r.h.s. [START_REF] Middleton | Measuring functional renormalization group fixed-point functions for pinned manifolds[END_REF][START_REF] Doussal | How to measure Functional RG fixed-point functions for dynamics and at depinning[END_REF].

In a real experiment, it is impossible to measure adiabatically, and instead one measures at a finite velocity,

∆ v (w -w ) := m 4 [w -u w ][w -u w ] c . (18)
By definition,

∆(w -w ) = lim v→0 ∆ v (w -w ). (19)
While ∆(w -w ) is the second cumulant of the effective action of the field theory [START_REF] Doussal | How to measure Functional RG fixed-point functions for dynamics and at depinning[END_REF], the expectation ( 18) is an observable. Perturbation theory allows us to calculate it as

∆ v (w) = ∞ 0 dt ∞ 0 dt ∆(w-vt+vt )R(t)R(t ), (20)
where R(t) is the response of the center of mass to an increase in w, and t R(t) = 1. (Usually, the response is defined w.r.t. an increase δF in force. Using δF = m 2 δw, the response w.r.t. to a force is normalized as t R(t) = 1/m 2 .) Eq. ( 18) implies that the integral of ∆ v (w) is independent of v. In general, both ∆(w) and R(t) may themselves depend on v. We show below in section III H that using the zerovelocity functions on the r.h.s. of Eq. ( 20) is sufficient at small driving velocities v, and the error made is probably O(v 3 ) or smaller, see Fig. 19.

An analytic expression for the amplitude of the rounding can be given by expanding Eq. ( 20) at w = 0 for small v,

∆ v (0) = ∞ 0 dt ∞ 0 dt ∆(0) + v∆ (0 + )|t -t | + O(v 2 ) R(t)R(t ) = ∆(0) + vτ ∆ (0 + ) + O(v 2 ), (21)
τ := ∞ 0 dt ∞ 0 dt |t -t |R(t)R(t ), (22)
τ := t ≡ ∞ 0 dt R(t)t. (23)
If R(t) decays exponentially, then τ = τ , which should remain a good approximation in most cases.

As an illustration for the operation defined in Eq. ( 20), consider ∆(w) = ∆(0)e -|w|/ξ , and

R(t) = τ -1 e -t/τ . (24)
Then, as plotted on Fig. 2,

∆ v (w) = ∆(0) e -|w|/ξ -τ v ξ e -|w|/(τ v) 1 -τ v ξ 2 . ( 25
)
This is a superposition of two exponentials, with the natural scales ξ and τ v. Since the cusp characteristic for depinning at the origin is rounded. This can be proven in general from Eq. [START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. II. Experiments[END_REF]. Note that ∆ v (w) is not analytic, as the expansion contains a term of order |w| 3 .

∆ v (0 + ) = 0, (26)
Since experiments work at a finite driving velocity, but we are mostly interested in the zero-velocity limit, the question arises of whether the folding of Eq. ( 20) can be undone. There are several possibilities to do this.

Boundary-layer analysis

As long as τ v ξ, the second term of Eq. ( 25) decays much faster than the first, allowing us to perform a boundarylayer analysis. This term was coined in the context of turbulence, where a turbulent bulk behavior has to be connected to a laminar boundary layer. There is a large mathematics and physics literature on the subject. Relevant keywords are boundary layer (physics literature) or singular perturbation theory (mathematics literature); a few references to start with are [START_REF] Wasow | Asymptotic expansions for ordinary differential equations[END_REF][START_REF] Bogolyubov | Encyclopedia of Mathematics: Perturbation Theory[END_REF][START_REF] Smith | Singular-Perturbation Theory[END_REF][START_REF] Hairer | Solving Ordinary Differential Equations II: Stiff and Differential-Algebra Problems[END_REF]. Let us proceed by noting that Eq. ( 20) can be approximated by the boundary-layer ansatz

∆ v (w) A v ∆ w 2 + δ 2 w , (27)
δ w = τ v, τ := t = ∞ 0 dt R(t)t , (28)
A v = ∞ 0 dw ∆(w) ∞ 0 dw ∆( w 2 + δ 2 w ) . ( 29
)
The amplitude A v ensures normalization. The quality of this approximation can be seen on Fig. 2: it works well for v small, but deteriorates for larger v.

We can use the boundary-layer formula [START_REF] Wiese | Functional renormalization for disordered systems: Basic recipes and gourmet dishes[END_REF] to plot the measured ∆ v (w) against w := w 2 + δ 2 w ; and then find the best δ w which removes the curvature of ∆ v (w). It yields δ w , and by extrapolation to w = 0 the full ∆(w). This idea is tested below in section III I.

Estimate of time scale

If in an experiment the response function is unavailable, using the boundary-layer ansatz [START_REF] Wiese | Functional renormalization for disordered systems: Basic recipes and gourmet dishes[END_REF], its characteristic time scale τ can be reconstructed approximatively from ∆ v (w) as

δ w = τ v lim w→0 ∆ (w) ∆ v (0) . (30)
In the numerator is written lim w→0 ∆ (w), which is obtained by extrapolating ∆ v (w) from outside the boundary layer, i.e w ≥ δ w = τ v, to w = 0.

Differential equation

First note that the response function R(t) defined in Eq. ( 24) satisfies the differential equation

(τ ∂ t + 1)R(t) = δ(t). (31)
Second, rewrite Eq. ( 20) as

∆ v v(t -t ) = t -∞ dt 1 t -∞ dt 2 ∆ v(t 1 -t 2 ) R(t -t 1 )R(t -t 2 ). (32)
Applying the differential operator of Eq. ( 31) both to t and t yields

(τ ∂ t + 1)(τ ∂ t + 1)∆ v v(t -t ) = ∆ v(t -t ) . (33)
In terms of the variable w, this relation can be simplified to

∆(w) = 1 -(τ v) 2 ∂ 2 w ∆ v (w). (34)
While Eq. ( 34) is more precise, and reconstructs ∆ v=0 (w) down to w = 0, the boundary-layer analysis may be more robust for noisy data. We will examine these procedures in section III I.

Other auto-correlation functions

Eq. ( 1) allows us to consider three different observables:

u w = u(t), ( 35
) uw = u(t), (36)
F w = F (u w ) = F u(t) . ( 37
)
What is measured in magnetic domain-wall experiments is the induced current, proportional to u(t) [START_REF] Urbach | Interface depinning, self-organized criticality, and the Barkhausen effect[END_REF][START_REF] Durin | Scaling exponents for Barkhausen avalanches in polycrystalline and amorphous ferromagnets[END_REF][START_REF] Kim | Direct observation of Barkhausen avalanche in Co thin films[END_REF][START_REF] Durin | The Barkhausen effect[END_REF]; in contact-line depinning where the interface is filmed, this is u(t) [START_REF] Le Doussal | Height fluctuations of a contact line: A direct measurement of the renormalized disorder correlator[END_REF]; finally, when stretching an elastic material as a knit, the force exerted on the system is measured by a piezo crystal [START_REF] Poincloux | Crackling dynamics in the mechanical response of knitted fabrics[END_REF]. This defines three auto-correlation functions

∆ v (w -w ) := m 4 [u w -w][u w -w )] c , (38)
∆ u(w -w ) := [ uw -v][ uw -v] c , (39)
∆ F (w -w ) := F w F w c . (40)
We will show in section III F that they satisfy

∆ F (w) = ∆ v (w) + ∆ u(w), (41)
∆ u(w) = - v 2 m 4 ∂ 2 w ∆ v (w). (42)

II. REVIEW OF KNOWN RESULTS FOR ABBM AND DPM

Key features for the ABBM model and the DPM are given on table I.

A. ABBM model

The response function is unchanged from the free theory

R(t) = m 2 e -m 2 t Θ(t). ( 43
)
The velocity distribution P t ( u) reads [START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. I. Theory[END_REF][START_REF] Bertotti | Hysteresis and Magnetism[END_REF][START_REF] Colaiori | Exactly solvable model of avalanches dynamics for Barkhausen crackling noise[END_REF][START_REF] Dobrinevski | Non-stationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model[END_REF]]

P t ( u) = m 2 m 2 u vm 2 -1 e -m 2 u Γ(vm 2 ) . ( 44
)
By construction it is normalized, its first moment is u = v, and

v ABBM m := u2 u = m -2 + v. (45)
The avalanche-size distribution P S δw (S), given a kick δw, reads [START_REF] Dobrinevski | Non-stationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model[END_REF]

P S δw (S) = m 2 δw e -m 4 (S-δw) 2 4S 2 √ πS 3/2 . ( 46
)
The avalanche-size exponent is τ = 3/2. The first moments are

1 = 1, S = δw, S ABBM m := S 2 2 S = m -4 + δw 2 .
(47) The duration distribution given a kick of size δw is

P T δw (T ) = exp - δwm 4 e T m 2 -1 δwm 6 [2 sinh(T m 2 /2)] 2 . ( 48
)
This distribution is normalized. The first moments to leading order in δw are

T = m 2 1 -γ E -ln(m 4 δw) + ... (49)
T 2 = π 2 3 δw + ... (50)
T 3 = 6ζ(3) δw m 2 + ... (51)
T 4 = 4π 4 15

δw m 4 + ... (52)
T ABBM m := T 3 3 T 2 = 6ζ(3) π 2 m 2 + ... = 0.730763 m 2 + ... ( 53
)
ABBM model Discretised Particle model (DPM) characteristic scale of effective (renormalized) force correlator

1 ρm = m 2 2 ln(m -2 ) -1 effective (renormalized) disorder ∆(w) ∆(0) -∆(w) = σ|w|, σ = 1 ∆(w) = m 4 ρ 2 m Li2(1 -e |w|/ρm ) + w 2 2ρ 2 m + π 2 6 critical force fc(m 2 ) = F (u = 0) fc(m 2 ) = 2 ln(m -2 ) response function R(t) = τ -1 e -t/τ τ -1 = m 2 τ -1 = τ -1 m := 2m 2 ln(m -2 )
avalanche-size distribution for infinitesimal kick P (S) ∼ S -τ e -S/Sm

τ = 3/2, Sm = m -4 τ = 0, Sm = ρm avalanche-duration distribution for infinitesimal kick P (T ) ∼ T -α e -T /Tm α = 2, Tm = τ = m -2 α = 0, Tm = τm = [2m 2 ln(m -2 )] -1 roughness exponent ζ, defined by u ∼ m -ζ ζ = 4 ζ = 2 -(2 reduced by logarithmic corrections) TABLE I.
Comparison of the ABBM model with the DPM.

B. The discretized particle model (DPM)

The discretized particle model (DPM) was introduced in Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF]. There the reader finds a thorough discussion of its quasi-static properties, encompassing all three main universality classes of extreme-value statistics: Gumbel, Weibull, and Fréchet.

Static quantities

As we assume forces to be distributed according to

P F (F ) = e -F 2 /2 √ 2π , (54)
this is the Gumbel class of Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF] with

A = 1/2, γ = 2, β(x) = x 2 /2, and β -1 (x) = √ 2x.
The name of this class stems from the fact that the blocking forces are distributed according to a Gumbel distribution ( [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF] Eq. (29), first line)

P G (a) = e -a Θ(a), (55)
f = √ 2 ln(m -2 ) -ln(a) (56) = f 0 c -ln(a) m 2 ρ m + ... (57)
The constant f 0 c , the scale ρ m , and the exponent ζ are

f 0 c = 2 ln(m -2 ) + O(m -1 ), ρ m = 1 m 2 2 ln(m -2 ) , ( 58
) ζ = 2 -. ( 59
) By ζ = 2 we mean ρ m ∼ m -ζ with ζ = 2.
The growth of Eq. ( 58) with 1/m is slightly slower, reduced by the logarithm in Eq. ( 58), and denoted ζ = 2 -. The effective disorder force-force correlator reads

∆(w) = m 4 ρ 2 m ∆(w/ρ m ), (60)
∆(w) = w 2 2 + Li 2 (1 -e w ) + π 2 6 . (61)
The avalanche-size distribution for an infinitesimal kick was obtained in Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF], where it was shown to be P (S) ∼ e -S/ρm . For a kick of size δw, this can be generalized to

P DPM δw (S) = 4δw sinh 1 2ρ m 2 e -S/ρm δw ρ 2 m e -S/ρm .
(62) Note that for the DPM, the avalanche size S is discrete. The normalization is constructed s.t. the first moment of this discrete measure is S = δw. This yields for the characteristic scale of avalanches

S m = S 2 2 S = 1 2 coth 1 2ρ m = ρ m + O(1/ρ m ) . (63)

Dynamic quantities

The DPM defined in Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF] advances instantaneously. The easiest way to endow it with a realistic dynamics is to consider the Langevin equation [START_REF] Le Doussal | Height fluctuations of a contact line: A direct measurement of the renormalized disorder correlator[END_REF]. If the disorder is needlelike as on the right of Fig. 1 (the original construction of [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF]), then either the particle is at rest blocked by a needle, or it moves, and the only force acting on it comes from the spring. Neglecting that the spring gets shorter during the movement, the response-function is given by R(t) ∼ P (S/v), where v = f c , resulting in

R DPM (t) = τ -1 m e -t/τm , τ m = ρ m f c = 1 2m 2 ln(m -2 )
.

(64) Stated differently, the velocity distribution is

P DPM ( u) = δ( u -f c ). ( 65
)
As a consequence, the distribution of durations T , given a kick δw, can be obtained from the avalanche-size distribution as

P DPM δw (T ) δw ρ m τ m e -T /τm . (66)

III. NUMERICAL RESULTS

A. Critical Force

First one integrates the equation of motion [START_REF] Le Doussal | Height fluctuations of a contact line: A direct measurement of the renormalized disorder correlator[END_REF], with forces as given by Eq. ( 6). This gives u(t), or u w as a function of the external point w = vt, defined s.t. u w = u(t). Example trajectories for u w -w, uw -v, and F w are plotted in Figs. 345.

It is important to note that the position u w and force F w are related, since Eq. (1) yields after averaging over time

∂ t u(t) = v = m 2 [w -u(t)] + F (t). (67)
Note that the overline, defined as an average over disorder realizations, can be performed as an average over time when driving the system at a finite velocity v > 0. For the discretised particle model, the critical force was computed analytically in Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF]. It is given by Eqs. ( 55)-( 56)

f c (m) := f G = √ 2 m -2
0 ln(m -2 ) -ln(a) e -a da .

(68) Expanding for small m, we find

f c (m) = f 0 c + γ E f 0 c - γ 2 E 2 + π 2 12 (f 0 c ) 3 + 2γ 3 E + γ E π 2 -2ψ (1) 4(f 0 c ) 5 + ... ( 69
)
f 0 c = 2 ln(m -2 ). (70)
In Fig. 6 we compare the measured critical force for the Ornstein-Uhlenbeck model, f OU c ≡ m 2 [w -u(w)] -v to the critical force (69) predicted by the particle model. We find that they agree, up to a constant

f OU c f c (m) -0.35. ( 71
)
This constant is not surprising, as the microscopic disorder of the Ornstein-Uhlenbeck process is different from the DPM, and the critical force is not universal. We note that the reported value 0.35 is almost the correlation e -1 ≈ 0.368 of the Ornstein-Uhlenbeck process at distance u = 1.

B. Velocity distribution

In the discretized-force model, and supposing that the forces remain constant between integers, the velocity distribution between site u and u + 1 is given by

P u ( u) = P F (f -u) G . (72)
Here P F (f ) is the initial force distribution (54), and the average is over the Gumbel distribution defined by Eqs. ( 55)-(56). As we are interested in the velocity distribution sampled equally in time and not in space, we still have to multiply with v/ u, resulting in

P t ( u) = v u P u ( u) (73) = v u 1 √ 2π m -2 0 e -u √ 2 - √ ln(m -2 )-ln(a) 2 e -a da.
This formula, evaluated numerically, is compared to simulations for m 2 = 10 -3 on Fig. 7. While the tail is correctly predicted, the amplitude for small u is underestimated. This is related to the underestimation of f c reported in Eq. ( 71). Indeed, we find a perfect fit for m 2 = 10 -3 under the replacement m -2 → 0.39m -2 , see Fig. 7.

C. The response function

Measurement prescription

The response function plays a key role as it enters into the rounding of the cusp, and we need to measure it to verify the prediction in Eq. (64). To this aim, we let the system relax to 73), the gray dotted one by the same formula, replacing m -2 → 0.39m -2 , yielding a perfect fit. Cyan dotted is the result for ABBM, Eq. ( 44). u = 0 and then kick it at time t = t kick , moving the center of the well from w to w + δw. The non-linear response function is then given by

R(t|δw) := u(t + t kick ) δw . (74)
The linear response is the limit of a small kick δw,

R(t) := lim δw→0 R(t|δw). (75)
In practice, δw can be chosen finite; we give suitable values on Figs. 8 and9. The response function is normalized, and its first moment defines a characteristic time scale τ ,

∞ 0 R(t) dt = 1, τ := t = ∞ 0 t R(t) dt. (76)
Comparing the measured time scale τ := t to the predicted one τ m from Eq. ( 64) is an important check of the theory.

Response function for DPM

We start with the response function for the discretised force model of Ref. [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF]. On Fig. 8 we show a numerical verification for the analytic prediction in Eq. (64). Already for a rather large mass of m 2 = 10 -2 the agreement between theory and simulation is rather good.

Response function for Ornstein-Uhlenbeck forces

On Fig. 9 we show the response function for Ornstein-Uhlenbeck forces. While for the DPM which has needlelike disorder, the response function on Fig. 8 had already converged to the asymptotic behavior of Eq. (64) for m 2 = 10 -2 , the convergence for Ornstein-Uhlenbeck forces is much slower, and one has to go to m 2 = 10 -5 to reach a similar agreement, albeit with a noticeable difference in the time scale τ = t . The rather slow convergence [45] is shown in the inset of Fig. 9. A second observation is that the response function R(t) for t → 0 starts at 1/τ ABBM = m 2 . This is a consequence of the continuity of F (u): Since in the beginning m 2 [w -u(t)] + F u(t) = 0, the response function is that of the free theory, itself equivalent to that of the ABBM model,

R(t) ≈ R free (t) ≡ R ABBM (t) = m 2 e -m 2 t Θ(t). (77)
The position u(t) then increases, and since pinning occurs at maxima of F (u), most likely F (u) decreases, leading to an increase in u as compared to the free theory. When u(t)u(t kick ) 1, we expect the forces F (u) to be decorrelated from its initial value. This estimates the boundary layer in u as δu ≈ 1. It is non-trivial to predict the boundary layer in t, i.e. the region in which R(t) rises, before decaying approximately as an exponential. After some experimentation, we found that this can be extracted by plotting the combination

δ BL (t) := (τ ∂ t + 1)R(t) (78)
appearing in Eq. ( 31), with τ defined in Eq. ( 76). By construction

∞ 0 dt R(t) = ∞ 0 dt δ BL (t) = 1. ( 79
)
If R(t) is exponentially decaying, δ BL (t) equals δ(t). Deviations lead to a smeared-out δ-function. On Fig. 10 we show δ BL (t) for several masses and time scales ranging from τ ≈ 14 to τ ≈ 596. Despite the enormous range of time scales, the resulting δ BL (t) is almost independent of τ , and decays to zero on a range of δt ≈ 2. We believe that this function could be extracted from extreme-value statistics, by considering the statistics of a Brownian motion close to one of its records. It is e.g. known that the fractal dimension of the record set, i.e. the position when the movement of the particle stops again, is 1/2 [START_REF] Mörters | Brownian Motion, Cambridge Series in Statistical and Probabilistic Mathematics[END_REF][START_REF] Benigni | Hausdorff dimension of the record set of a fractional Brownian motion[END_REF].

The response function at a finite driving velocity

The response function at a finite driving velocity and the associated time scales are defined as

R v (t) := lim δw→0 1 δw u(t) δw v -v , (80)
t v = ∞ 0 dt R v (t)t. ( 81
)
The expectation value on the r.h.s. of Eq. ( 80) is the expectation of u(t), given a driving velocity v, and an additional kick at time t = 0; we have subtracted the expectation without a kick, u(t)

δw=0 v = v.
The time scale t v defined in Eq. ( 81) should then behave as

1 t v f c -v, v small. ( 82
)
What appears on the r.h.s. is the relative velocity between the "ballistically" moving particle as given by Eq. ( 65) and the advancement of the confining potential. On the other hand, at large driving velocity, the disorder acts as a thermal noise, and the response function reduces to that of the free theory, equivalent to that of the ABBM model, resulting in

1 t v τ -1 ABBM = m 2 , v large. ( 83
)
v = 0 (red), v = 0.3 (blue), v = 0.5 (green), v = 1 (violet), v = 1.5 (brown), v = 2 (
black) and v = 3 ≈ fc (magenta). Inset: Scaling collapse for the velocities v ≤ 2, using the measured time scale t instead of τm for the rescaling. (DNS, δu = 10 -6 , δt = 10 -4 , N = 10 7 ). FIG. 13. Avalanche-size distribution P (S) for a particle with forces given by Eq. ( 6). The theoretical curves are the kicked ABBM model as given by Eq. ( 46) (cyan dotted), and the DPM as given by Eq. ( 62 These two curves are plotted on Fig. 11. They intersect at v = f c -m 2 . Intuitively we expect the transition to take place at v = f c , and to be smoothed out due to the finite width of the velocity distribution (73). As we can see on Fig. 11, the transition gets sharper when m 2 decreases. A phase transition at v = f c is expected in the limit of m 2 → 0.

Our final observation is that the time scale t v defined in Eq. ( 82) can be used for a scaling collapse of the response functions R v (t) defined in Eq. ( 80), see Fig. 12. Details of the analysis are given in Ref. [45].

D. Avalanche-size distribution

The avalanche-size distribution is shown on Fig. 13, for m 2 = 10 -4 , overlayed with two theoretical curves: the kicked ABBM model as given in Eq. ( 46) (bright dotted cyan), and the DPM given in Eq. (62) (dark blue dashed).

For small S, one sees the S -3/2 behavior characteristic for ABBM coincide with the theoretical curve (46) (cyan, dashed). For large S, this crosses over to the prediction of Eq. (62) for the DPM with avalanche exponent τ = 0 (blue, dashed). Note that there are no adjustable parameters, and both curves respect their own normalization. The crossover takes place at S ≈ 1, and extends over about half a decade in both directions.

E. Avalanche-duration distribution

The avalanche-duration distribution is shown on Fig. 14, superimposed with the two theoretical predictions [START_REF] Dobrinevski | Field theory of disordered systemsavalanches of an elastic interface in a random medium[END_REF] for ABBM (cyan dotted), and (66) for the DPM (blue, dashed). The simulations agree very well with ABBM for small times, while the overall weight for large times is seemingly overestimated in Eq. (66), contrary to what one saw on Fig. 13 for the avalanche-size distribution. Normalization issues are more . This is compared to the theory prediction [START_REF] Dobrinevski | Field theory of disordered systemsavalanches of an elastic interface in a random medium[END_REF] for ABBM (cyan dotted), and (66) for DPM (blue, dashed).

pronounced since the singularity for small times P (T ) ∼ T -2 is stronger than the P (S) ∼ S -3/2 for the avalanche size. Another reason for the slow convergence is the relatively broad avalanche-velocity distribution, assumed to be restricted to u = f c in the derivation of Eq. ( 48).

F. Correlators ∆v(w), ∆ u(w), ∆F (w)

In section I B 5, we had defined the three correlation functions ∆ v (w), ∆ u(w), and ∆ F (w), which are the correlations of three forces: confining well, friction, and disorder. Since they add up to zero, two variables are independent, and one would expect three independent correlation functions, which are most symmetrically expressed as ∆ v (w), ∆ u(w), and ∆ F (w). This expectation is incorrect: there are only two independent quantities, summarized in the relation

∆ v (w) + ∆ u(w) = ∆ F (w). (84)
A numerical verification is presented on Fig. 15. An analytical proof can be given too: identifying w ≡ vt, one has

∆ F (w -w ) = F w F w c = [ uw + m 2 (u w -w)][ uw + m 2 (u w -w )] c = uw uw c + m 4 (u w -w)(u w -w ) c + m 2 uw (u w -w ) + (u w -w) uw c . ( 85
)
The last term can be written as

uw (u w -w ) + (u w -w) uw c = (∂ t + ∂ t ) (u w -w)(u w -w ) c = 1 v (∂ w + ∂ w ) ∆ v (w -w ) = 0. (86)
This proves Eq. (84). On Fig. 16, we find numerically satisfied another relation, This relation expresses the time derivative of u(t) = u w by its dependence on w = vt: ∂ t u(t) = v∂ w u w . Thus knowing ∆ v (w) is enough, and the two other quantities can be expressed as

m 4 ∆ u(w) = -v 2 ∂ 2 w ∆ v (w) . ( 87
)
∆ u(w) = - v 2 m 4 ∆ v (w), (88)
∆ F (w) = ∆ v (w) - v 2 m 4 ∆ v (w). (89)
Knowing ∆ v (w) it is easy to find ∆ u(w) by taking two derivatives. It is already less obvious to reconstruct ∆ v (w) from ∆ u(w), as this procedure involves two integration constants.

In principle the latter are fixed since all correlations vanish for w → ∞; in practice, however, fluctuations grow with increasing w, and they show up in the two integration constants.

It is even more involved to reconstruct ∆ v (w) from ∆ F (w): Formally, this can be achieved by the series

∆ v (w) = ∞ n=0 v m 2 d dw 2n ∆ F (w). ( 90
)
As it is rather badly converging, it may well be useless in practice.

G. Measuring ∆(w) at vanishing driving velocity

In an experiment, it is difficult to measure ∆(w) = lim v→0 ∆ v (w). In our simulation, we can do this: We move the parabola from w → w + δw, and then wait until the dynamics cedes. Due to Middleton's theorem [START_REF] Aa | Asymptotic uniqueness of the sliding state for charge-density waves[END_REF], the position u w is history independent. From u w we obtain ∆(w) via formula [START_REF] Csikor | Dislocation avalanches, strain bursts, and the problem of plastic forming at the micrometer scale[END_REF]. The DPM [START_REF] Doussal | Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records[END_REF] predicts that

∆(w) = m 4 ρ 2 m ∆(w/ρ m ), (91)
∆(w) = Li 2 (1 -e |w| ) + w 2 2 + π 2 6 . (92)
To check this relation, we study the experimentally measured ∆ exp (w), and then invert Eq. (91) to obtain

∆exp (w) = 1 m 4 ρ 2 m ∆ exp (w/ρ m ). (93)
Figure 17 shows convergence against the theoretical result.

H. The disorder correlator ∆v(w) at a finite driving velocity

For a finite driving velocity, Eq. ( 20) predicts that the measured ∆ v (w) is obtained from the zero-velocity correlator FIG. 18. The numerically measured ∆v(w) for v = 0.5 (red, dashed, with 95%-confidence intervals in shaded red) as defined in Eq. ( 18). For w > 1000 it agrees well with the theoretical prediction (91) (black, solid). The inset shows the same curves, superimposed with the theoretical prediction (91) folded according to Eq. ( 20) with the numerically measured response function (green), lying underneath ∆v(w) obtained numerically (red, dashed).

Δ( ) -Δ ( )

FIG. 19.

The measured amplitude of the rounding v -1 [∆v=0(0) -∆v(0)] for m 2 = 0.01 and v = 0.1, 0.3, 0.5, 1, 2, 3.5, 4; fc 3.035. Error bars (green) are included in the fit and represent 99% confidence intervals on the data. We compare several curves. In black is shown the prediction from Eq. ( 22) with τ = τm, and ∆ (0 + ) as given by Eq. (91). In blue the prediction from Eq. ( 22) with τ = t , and ∆ (0 + ) as measured in the simulation, accompanied by error bars in green. In red a weighted fit to the velocities v ≤ 0.5, verifying Eq. ( 22) with a relative precision of 3 × 10 -3 . ∆(w) by folding with two response functions. In the inset of Fig. 18 we check that this construction works for v = 0.5, by folding the numerically measured zero-velocity correlator ∆(w) with the numerically measured zero-velocity response function R(t).

We numerically established that the response function evolves with v from their zero-velocity limit (see Fig. 12), and the same may happen for the effective disorder-correlator, technically the effective action of the theory. While the agree-parameters τ := t tR(t) τ from boundary layer analysis via Eq. ( 27)

τ from differential Eq. ( 34)

τ from Eq. ( 30) 27) is the most precise, followed by the boundary-layer analysis using Eq. ( 34), whereas the approximation (30) has a relatively large error. In sections I B 2 to I B 4 we proposed two procedures to reconstruct the zero-velocity correlator ∆(w), based either on a boundary-layer analysis (section I B 2), or a differential equation (section I B 4). These methods contain a parameter, the time-scale τ of the response function, in the combination δ w = τ v, and allow us to extract the latter if unknown. Let us analyze these methods in turn.

m 2 = 0.01, v = 0.

Boundary-layer analysis

As proposed in section I B 2, we use the boundary-layer formula [START_REF] Wiese | Functional renormalization for disordered systems: Basic recipes and gourmet dishes[END_REF] to plot on Fig. 20 the measured ∆ v (w) against w = w 2 + δ 2 w , δ w = vτ , using the timescale τ = t measured from the response function. By extrapolation to w = 0 we obtain the full ∆(w). This works decently well, especially for the smaller driving velocity v = 0.1.

Differential equation

A second unfolding procedure was proposed in section I B 4. Repeating the analysis performed on Fig. 20, we show on Fig. 21 the reconstructed ∆(w). For v = 0.1 the unfolding procedure reproduces the v = 0 correlator with high precision. The agreement is not as good for v = 0.3.

Reconstructing the time scale τ

In the last two sections, we tested the reconstruction procedures under the assumption that the time scale τ was known from an independent measurement of the response function. In practice, τ may not be available. In that case, we try to find δ w = vτ , which best removes the curvature of ∆ v (w). The estimated values for τ are reported on table II, together with an estimate of τ from Eq. [START_REF] Rosso | Numerical calculation of the functional renormalization group fixed-point functions at the depinning transition[END_REF]. While the latter gives only fair results, both reconstruction procedures allow us to extract the time scale τ rather precisely.

IV. SUMMARY AND CONCLUSION

We have shown that there is more to mean-field theory than replacing the effective force landscape by a random walk (ABBM model). As forces cannot grow unboundedly, they must finally saturate, leading to a different regime of uncorrelated forces (DPM). Above we propose to describe the crossover by modeling forces as an Ornstein-Uhlenbeck process. Using numerical simulations supported by analytical results for all key ingredients, we quantify the signatures expected in experiments. The key observable is the effective force-force correlator, which is readily accessible in experiments. As experiments necessitate a finite driving velocity, the measured signal is always smeared out. Above we developed and tested procedures to reconstruct the zero-velocity response and force-force correlations from finite-velocity measurements. We have already tested our procedure with success for magnetic domain walls (collaboration with G. Durin) and knits (collaboration with A. Douin and F. Lechenault). Each of these systems has its own peculiarities, on which we will report in forthcoming publications.

FIG. 2 .

 2 FIG. 2. ∆(w) = e -w (blue solid), and the finite-velocity correlator (25) (red) for τ v = 0.3, compared to the boundary-layer approximation (27) (green dotted).

FIG. 3 .

 3 FIG. 3. uw -w for δt = 10 -4 , v = 0.1, m 2 = 0.1, INS. See section B for implementation details. Red points are data points equally spaced in time, the green line between them is an interpolation.

FIG. 4 .

 4 FIG.4. u(w) -v with the same parameters as in Fig.3.

FIG. 6 .

 6 FIG.6. The critical force fc as a function of m. Thick black: the results of the integral (68). The small-m expansion is in blue, cyan, green, and yellow. The data points are from our numerical simulation. DNS, δu = 10 -2 , δt = 10 -4 , N = 3 × 10 6 .

FIG. 7 .

 7 FIG. 7. P ( u) for m 2 = 10 -3 , δt = 10 -3 , v = 0.1, N = 5 × 10 8 , DNI (orange, solid line). The blue dashed curve is given by Eq. (73), the gray dotted one by the same formula, replacing m -2 → 0.39m -2 , yielding a perfect fit. Cyan dotted is the result for ABBM, Eq. (44).

FIG. 8 .

 8 FIG. 8. The response function for the discretized force model (needle disorder) for m 2 = 10 -2 (blue). The red dashed curve is an exponential function with time scale τm as given by Eq. (64). Inset: ibid for ln R(t). DNS, δt = 10 -4 , δw = 0.1.

FIG. 9 .

 9 FIG. 9. The response function for the Ornstein-Uhlenbeck process: analytical result in black dashed. Numerical results from bottom to top are for m 2 = 10 -2 (blue), m 2 = 10 -3 (orange), m 2 = 10 -4 (cyan), and m 2 = 10 -5 (red). DNS, δt = 10 -4 , N = 10 8 . The two larger masses have δw = 0.1 and δu = 10 -6 , the two smaller ones δw = 1, and δu = 10 -4 . Inset: The measured timescale t (dots) compared to the prediction for τm in Eq. (64).

6 τ

 6 FIG.10. The function δBL(t) := (τ ∂t + 1)R(t) for the measured response function with m 2 = 10 -2 (blue), m 2 = 10 -3 (orange), m 2 = 10 -4 (cyan). Parameters as in Fig.9. ***plots missing, one could also put δBL(t) on the y-axis.

m 2  10 - 4 FIG. 11 .

 210411 FIG. 11. Inverse of the measured time scale t v := ∞ 0 dt tRv(t), as a function of v. The data points are given with 99% confidence intervals (green). Superimposed are the two analytical curves (81), valid for v ≤ fc, and (82), valid for v ≥ fc. DNS, δt = 10 -4 , δu = 10 -6 for m 2 = 0.01, and δu = 10 -4 for m 2 = 10 -4 .

FIG. 12 .

 12 FIG.12. Main plot: The response function rescaled with the theoretically predicted time scale τm. Velocities from top to bottom are v = 0 (red), v = 0.3 (blue), v = 0.5 (green), v = 1 (violet), v = 1.5 (brown), v = 2 (black) and v = 3 ≈ fc (magenta). Inset: Scaling collapse for the velocities v ≤ 2, using the measured time scale t instead of τm for the rescaling. (DNS, δu = 10 -6 , δt = 10 -4 , N = 10 7 ).

  FIG.13. Avalanche-size distribution P (S) for a particle with forces given by Eq. (6). The theoretical curves are the kicked ABBM model as given by Eq. (46) (cyan dotted), and the DPM as given by Eq. (62) (blue, dashed). INS, m 2 = 10 -4 , δw = 1, δt = 10 -4 , Sm := S 2 /(2 S ) = 2408.89, ρm = 2329.95, N = 10 8 .

FIG. 14 .

 14 FIG.[START_REF] Burridge | Model and theoretical seismicity[END_REF]. P (T ) measured numerically via INS for m 2 = 10 -3 , δw = 1, δt = 10 -3 , N = 5 × 10 8 . This is compared to the theory prediction[START_REF] Dobrinevski | Field theory of disordered systemsavalanches of an elastic interface in a random medium[END_REF] for ABBM (cyan dotted), and (66) for DPM (blue, dashed).

Δ

  FIG. 15. The measured ∆v (cyan) , ∆F (orange, dashed) and ∆ u (purple, dotted) for m 2 = 0.01 at v = 0.5. The inset shows that the difference ∆F -∆v -∆ u vanishes. (DNS, δu = 0.01, δt = 10 -4 , N = 10 8 ).

v 2 Δ v ′′ (w) m 4 +

 24 FIG. 16. Numerical test of Eq. (88) for m 2 = 10 -2 , v = 0.5. Plotted is ∆ u(w) (cyan, dottedd) and -v 2 m 4 ∆ v (w) (red, dashed). The inset shows the difference. (DNS, δu = 0.01, δt = 10 -4 , N = 10 8 ).

FIG. 17 .

 17 FIG. 17. Convergence of the measured ∆(w) for m 2 = 10 -2 (gray, dot-dashed), m 2 = 10 -3 (cyan, dotted) and m 2 = 10 -4 (red, dashed). This is compared to three theoretical curves: an exponential function with the same slope at w = 0 (green, top curve), Eq. (92) (black, middle curve), and the 1-loop FRG-result (blue, bottom curve), see Eqs. (4.5) and (4.11) of[START_REF] Le Doussal | 2-loop functional renormalization group analysis of the depinning transition[END_REF]. The inset shows the difference of the measured ∆(w) at m 2 = 10 -4 , and the theoretical curves in their respective colors, together with error bars at a 99% confidence level.

FIG. 20 .

 20 FIG.[START_REF] Alessandro | Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. II. Experiments[END_REF]. Measurement of ∆v(w) for v = 0 (solid blue, top), v = 0.1 (solid red, middle), and v = 0.3 (solid green, bottom). We show unfolding via a boundary layer analysis, using τ := t = 14.08 as measured for the response function, both for v = 0.1 (black dots), prolonged to zero (red, dashed) and v = 0.3 (black dots), prolonged to zero (green, dashed). The variable w = w 2 + (τ v) 2 . DNS, δt = 10 -4 , δu = 10 -2 , N = 10 8 .

m 4 FIG. 21 .

 421 FIG.[START_REF] Vergne | Quelques aspects statistiques des processus d'aimantation dans les corps ferromagnétiques. Cas du déplacement d'une seule paroi de Bloch à 180 • dans un milieu monocristallin aléatoirement perturbé[END_REF]. Measurement of ∆v(w) for v = 0 (solid blue, top), v = 0.1 (solid red, middle), and v = 0.3 (solid green, bottom). We show unfolding via the differential equation (34) using τ := t = 14.08 as measured for the response function, both for v = 0.1 (red, dashed) and v = 0.3 (green, dotted). DNS, δt = 10 -4 , δu = 10 -2 , N = 10 8 .

TABLE II .

 II Comparison of the measured time scale τ , and its estimation using the methods of section I B 2-I B 4. The estimation via the differential equation (

	1 14.08	14.10	14.08	14.87
	m 2 = 0.01, v = 0.3 14.08	14.08	14.08	13.90
	m 2 = 10 -4 , v = 0.5 596	587	587	664

ACKNOWLEDGMENTS

It is a pleasure to thank A. Kolton, G. Mukerjee, A. Rosso and B. Walter for fruitful discussions. The questions raised here are inspired by our ongoing collaboration with experimentalists, and we are grateful for the inspirations brought to us by G. Durin (magnetic domain walls), as well as A. Douin and F. Lechenault (knitting).

Acknowledgments 13

A. Correlations of an Ornstein Uhlenbck process 13 B. Numerical implementations

Appendix A: Correlations of an Ornstein Uhlenbck process Suppose that

This equation is solved by

du 1 e -(u-u1) ξ(u 1 ).

(A2)

It leads to microscopic correlations

Appendix B: Numerical implementations

We used several numerical implementations: (i) Direct numerical simulation (DNS). To solve the coupled set of differential equations ( 1)-( 3) we use a space discretization δu = 10 -6 to 10 -2 (depending on m) to first obtain the random forces F (u) for u = nδu, n ∈ N. F (u) is then linearly interpolated between these points. We finally solve Eq. ( 1) with the Euler method, using a time-discretization of δt = 10 -4 .

(ii) Improved numerical solver (INS). In the coupled equations of motion ( 1)-( 3), the force F(t) := F u(t) can statistically equivalently be modeled as [START_REF] Dornic | Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions[END_REF][START_REF] Dobrinevski | Field theory of disordered systemsavalanches of an elastic interface in a random medium[END_REF][START_REF] Kolton | Distribution of velocities in an avalanche, and related quantities: Theory and numer-References 13[END_REF]

The advantage of this scheme is that there is only one parameter, namely the time discretization δt, but none for the space discretization δu. The effective space discretization is δu ≈ u(t) δt, thus is finer when the system moves slowly. Treating the multiplicative noise of Eq. (B1) however demands some care [50]. We use both a direct simulation with a very small time-step as the scheme prosed in [START_REF] Dornic | Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions[END_REF] and tested in [START_REF] Dobrinevski | Field theory of disordered systemsavalanches of an elastic interface in a random medium[END_REF][START_REF] Kolton | Distribution of velocities in an avalanche, and related quantities: Theory and numer-References 13[END_REF].

The number of samples is denoted