Avalanche-size distribution at the depinning transition: A numerical test of the theory

Avalanche-size distribution at the depinning transition:
A numerical test of the theory

Alberto Rosso LPTMS; CNRS and Universite Paris-Sud, UMR 8626, 91405 Orsay Cedex, France. CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France.    Pierre Le Doussal CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France.    Kay Jörg Wiese CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France.
August 1, 2019
Abstract

We calculate numerically the sizes of jumps (avalanches) between successively pinned configurations of an elastic line () or interface (), pulled by a spring of (small) strength in a random-field landscape. We obtain strong evidence that the size distribution, away from the small-scale cutoff, takes the form where is the scale of avalanches, and the roughness exponent at the depinning transition. Measurement of the scaling function is compared with the predictions from a recent Functional RG (FRG) calculation, both at mean-field and one-loop level. The avalanche-size exponent is found in good agreement with the conjecture , recently confirmed to one loop via the FRG. The function exhibits a shoulder and a stretched exponential decay at large , , with in . The function , universal ratios of moments, and the generating function are found in excellent agreement with the one-loop FRG predictions. The distribution of local avalanche sizes , i.e. of the jumps of a subspace of the manifold of dimension , is also computed and compared to our FRG predictions, and to the conjecture .

I Introduction

Elastic objects pinned by a random substrate are ubiquitous in nature. The competition between elastic restoring forces and quenched disorder results in multiple metastable states. Upon applying an external force one observes collective jerky motion which proceeds by sudden jumps, called avalanches. Examples are the Barkhausen noise in magnets UrbachMadisonMarkert1995 (); MehtaMillsDahmenSethna2002 (); DahmenSethnaKuntzPerkovic2001 (); CarpenterDahmenSethnaFriedmanLoverdeVanderveld2001 (); DahmenSethnaPerkovic2000 (); PerkovicDahmenSethna1995 (); DahmenKarthaKrumhanslRobertsSethnaShore1994 (), jumps in the creep motion of magnetic domain walls LemerleFerreChappertMathetGiamarchiLeDoussal1998 (); RepainBauerJametFerreMouginChappertBernas2004 (); MetaxasJametMouginCormierFerreBaltzRodmacqDienyStamps2007 (); SethnaDahmenMyers2001 (), avalanches in the depinning of a contact-line of a fluid PrevostRolleyGuthmann2002 (); PrevostRolleyGuthmann1999 (); MoulinetGuthmannRolley2002 (); MoulinetRossoKrauthRolley2004 (), or in dislocation and crack propagation MorettiMiguelZaiserMoretti2004 (); PonsonBonamyBouchaud2006 (); Ponson2007 (); BonamySantucciPonson2008 (), and stick-slip motion of e.g. tectonic plates, responsible for earthquakes FisherDahmenRamanathanBen-Zion1997 (); DSFisher1998 (); SchwarzFisher2003 (); JaglaKolton2009 (). Avalanches have also been studied in models without quenched substrate disorder, such as in sandpile models and in granular matter TangBak1988 (); DharRamaswamy1989 (); BanerjeeSantraBose1995 (); Dhar2006 (). An important characteristics of avalanche motion is its scale invariance, self-organized criticality, and a broad distribution of the sizes of avalanches, for sizes between a small- and large-scale cutoff . Pinned elastic manifolds are an important prototype of a much wider class of phenomena, reaching far outside physics, e.g. into economy and finance, where extreme (and sometimes catastrophic) events are sufficiently frequent and large to dominate most observables. In this context, it is clearly of importance to understand how the avalanche-size probability is cut off at the large scales, for .

Although avalanche motion of pinned manifolds has been studied for a while in numerics NarayanMiddleton1994 (); LacombeZapperiHerrmann2001 (); TanguyGounelleRoux1998 (), most work focused on measuring the avalanche-size exponent , with minimal guidance from the theory. This is mainly because no analytic approach was available besides mean-field and scaling arguments. The most notable one was proposed by Narayan and Fisher (NF) NarayanDSFisher1993a () on the basis of the unproved assumption that the avalanche density remains finite at the depinning threshold, resulting into

(1)

Here is the roughness exponent at the depinning transition. Progress both in constructing the field theory of the depinning transition ChauveLeDoussalWiese2000a (); LeDoussalWieseChauve2002 (); LeDoussalWiese2003a (); FedorenkoLeDoussalWiese2006 () following the pioneering work on the Functional RG (FRG)NattermanStepanowTangLeschhorn1992 (); NarayanDSFisher1993a (); LeschhornNattermannStepanow1996 () and in developing new powerful algorithms RossoKrauth2001b (); RossoKrauth2002 (); RossoHartmannKrauth2002 (); RossoKrauthLeDoussalVannimenusWiese2003 (); MoulinetRossoKrauthRolley2004 (); BolechRosso2004 (); KoltonRossoGiamarchi2005 () had focused mostly on structural properties of the pinned manifold, such as the precise determination of . Even an appropriate definition of static and dynamic avalanches, allowing contact with the field theory, had remained unclear. It was given in the statics LeDoussal2006b (); MiddletonLeDoussalWiese2006 (); LeDoussal2008 () and at depinning LeDoussalWiese2006a (); RossoLeDoussalWiese2006a () using a confining quadratic potential; it led to the measurement, with great accuracy, of the renormalized disorder correlator , i.e. the fixed point of the FRG. Only very recently we succeeded in computing the distribution of avalanche sizes within the FRG LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c (); FedorenkoLeDoussalWieseInPrep (). The calculation at tree level gave mean-field predictions (some of them new and non-trivial), valid above the upper critical dimension , i.e. for . The one-loop calculation gave an expansion to order , with . Remarkably, the conjecture (1) was confirmed to accuracy. It is thus of great interest to test these predictions in numerics.

The aim of the present paper is to compute numerically the jumps (avalanches) between successively pinned configurations of an elastic line () and interface (). The convenient setting to compare with the recent predictions from the FRG is to submit the manifold to an external quadratic well, i.e. a spring. We will study mostly random-field disorder, but we also check that the results are the same for random-bond disorder, as is predicted at depinning and was checked in our previous work RossoLeDoussalWiese2006a () for the renormalized disorder correlator . Most of the numerical method is similar to our previous work RossoLeDoussalWiese2006a ().

The outline of this article is as follows: We define in section II the model and numerical procedure; and in section III an avalanche, its size, the characteristic scales and the scaling functions. In section IV the reader will find our numerical results for the avalanche-size distribution, and their comparison to our analytical results in and . In section V we compare our numerical and analytical results for the universal ratios of algebraic moments, the . In section VI we do the same for the generating function of exponential moments, i.e. the characteristic function of the size distribution, denoted , in and . Finally, in Section (VII), we compute the distribution of local avalanches in and compare with the predictions.

Ii Numerical procedure: parabola and metastable states

Let us now describe the model and algorithm, in for simplicity. The procedure is very similar to our previous work RossoLeDoussalWiese2006a (). The interface is discretized as , , and periodic boundary conditions are taken: , . We start from a flat interface () embedded in a parabolic potential. The equation of motion is

(2)

is the disorder force. We distinguish two different microscopic disorders:

(i) random force (RF): for each integer value of we take a random number extracted from a normal distribution. The value of the random force for non-integer values of is given by the linear interpolation of the forces at the two closest integers . Forces for different are independent.

(ii) random bond (RB): the random force is derived from a random potential: . For each integer value of , the potential is a random number normally distributed. The interpolation of is done by means of a cubic spline connecting random numbers. Two extra conditions are needed in order to define a spline: we have taken and . In our simulations . When the line advances beyond , a new spline, with new random numbers is generated. Potentials for different are independent.

The value is the center of mass of a confining potential for each point , of the form . In the simulation, is increased from . For each value of a metastable state is computed. Increasing , a stationary sequence of metastable states (independent of the initial configuration) is reached, as observed in Ref. RossoLeDoussalWiese2006a (). This is the steady state on which we focus. Our main results concern an elastic string in of size with RF disorder, but we have also studied RB disorder, see Fig. 1, and a -dimensional elastic interface of size with periodic boundary conditions. As expected, for the depinning transition, the RB case falls in the same universality class as the RF case and results are very similar.

Iii Definitions and observables

For given the manifold moves to a metastable state , i.e. a state dynamically stable to infinitesimally small deformations. Following the notation of LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c (), we define the center of mass of the metastable configuration

(3)

with the linear size of the system (number of points), and the dimension. One then increases , and a smooth forward deformation of results (for smooth short-scale disorder) while the state remains stable. At some the state becomes unstable and the manifold, for moves until it is blocked again in a new metastable state (also locally stable). This process is called an avalanche and its size is defined as the area swept by the line as it jumps between the two consecutive metastable states:

(4)

The distribution of avalanche sizes is expected to exhibit universality, i.e. independence of short scales, for sizes . The short-scale cutoff corresponds to the area spanned by a single monomer on the scale of the discretization of the disorder (in our units ). In the limit a critical point is reached, resulting in a power-law distribution of avalanche sizes. To properly define the problem, including the stationary measure, it is essential to consider a small . Then, the internal correlation length is finite: it can e.g. be measured from the structure factor leading to RossoLeDoussalWiese2006a ()

(5)

is large in the small- regime considered here. As a result, the distribution of avalanche sizes is cut off by the large scale , defined as

(6)

It is expected to scale as at small . Here and below we define the (normalized) distribution of avalanche sizes , as well as its moments

(7)

from the sequence of measured avalanches , .

The scale is important as it allows to define universal functions. In the variable the avalanche-size distribution should become universal. Indeed, one of the predictions of the FRG theory is that if the exponent satisfies which is the case here, then the distribution of avalanche sizes for takes the form as , i.e. ,

(8)

The function is universal and depends only on the space dimension . Note that the normalized probability depends on the cut-off via the first moment which cannot be predicted by the theory, hence is an input from the numerics. It is important to stress that while the function is universal and convenient for data analysis, it is not a probability distribution and is not normalized to unity. Rather, it satisfies from its definition (8) and using (6) the two normalization conditions

(9)
(10)

Here and below we use the notation to denote an integration over and distinguish it from a true expectation value over , denoted .

Iv The avalanche-size distribution

Figure 1: Random Field and Random Bond (RB) (). A fit with a power law gives the exponent . The agreement with Eq. (12) is discussed in the text.

The rescaled avalanche-size distribution can be written as

(11)

where is the avalanche-size exponent, and the universal cutoff function 111by universal we mean w.r.t. short scale details. Of course this function is characteristic of a large scale cutoff provided by a parabolic well, which tends to a constant for . For the present model, the only analytical prediction prior to our work LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c () concerns the exponent , via the above mentioned NF NarayanDSFisher1993a (); NarayanDSFisher1993b () conjecture

(12)

where is the roughness exponent at the depinning transition. Exact solution DSFisher1998 () of a mean field toy model of avalanches, which turns out to be related to the famous Galton process WatsonGalton1875 () in genealogy, gives an exponent . This exponent is also the one expected if we replace in the NF conjecture. This does however not constitute a first-principle calculation starting from the model of the pinned interface. The latter was only possible using the FRG LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c (). The summation of all tree diagrams within the FRG is shown to be asymptotically exact for and leads to the mean-field prediction LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c () for and for the full rescaled avalanche-size distribution (see below).

We now discuss our numerical results starting with the avalanche-size exponent . Note that the data in Fig. 1 contain both random-field and random-bond disorder and that, as expected from the universality of the depinning fixed point, the results are indistinguishable. Hence in the following we focus on RF disorder. For , a direct power-law fit of our numerical data (see Fig. 1) gives

(13)
Figure 2: Numerical extrapolation of the exponent to mass . We find

This value has to be compared with the conjecture of Eq. (12). The roughness exponent is known numerically with a good accuracy from system sizes () and , as RossoHartmannKrauth2002 (). This value for gives . Hence the estimate (13) is slightly smaller than the value of obtained from the conjecture. There are several possible explanations for this.

First one notes that although (13) is extracted from pieces of which have already well converged in terms of and , the resulting window of sizes is limited. Although we took this into account in estimating (13), we cannot exclude a further small upward shift in the central value as the window size increases.

Second, we have also measured the effective exponent for the sizes and masses used here. From measurements of we extract as can be seen on Fig. 2. We have checked that comparable estimates can be extracted from the structure factor , as measured also in RossoLeDoussalWiese2006a (), using fits taking into account the mass. Inserting this value for an effective into Eq. (12), this results in an effective value for , which is in much better agreement with our measured value (13).

Finally, deviations from the conjecture for the asymptotic value of are still, strictly speaking, possible, but if they exist they must be around or below the error of in Eq. (13). This does not rule them out since, as discussed in LeDoussalWiese2008c (), if present they are expected to be small 222Note that the exact solution in LeDoussalWiese2008a () shows that the conjecture is valid for RF disorder with sufficiently LR correlations (e.g.  and for a Brownian force landscape), but fails for RF disorder with SR correlations, e.g.  and , i.e. for the Gumbel class with . .

Within the FRG LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c () it is possible to compute the universal scaling function. For summation of all tree diagrams gives

(14)

The one-loop FRG calculation gives

(15)

with exponents

(16)
(17)

where and for the RF class, relevant to the present study. The constants , and depend on , and must satisfy the normalization conditions (9), (10). At first order in they are , , , . As usual, the one-loop results for the exponents , and for the parameters , , and are exact up to .

Figure 3: Random Field (). Blow up of the power-law region. The red solid curve is given by Eq.(14) , the black dashed line by Eq.(15), with , and .
Figure 4: Random Field (). Blow up of the tail region. The red solid curve is given by Eq.(14) , the black dashed line by Eq.(15), with , and .
Figure 3: Random Field (). Blow up of the power-law region. The red solid curve is given by Eq.(14) , the black dashed line by Eq.(15), with , and .

To analyze our numerical data for the avalanche-size distribution, we have first computed and from the data, which allowed to determine numerically the universal (and parameter-free) function using (6) and (8). Hence by construction the numerical data satisfy conditions (9) and (10). They are plotted in Figs. 4 and 4, with emphasis either on the power-law region or on the tail. Note that for the different values of and used here, the data have converged, with the exception of the last point for very large avalanches over-suppressed by the finite size of the interface in the smallest samples, and of the region of very small avalanches, which are cut off at .

To compare the numerical data with the mean-field and one-loop predictions, we use two procedures:

In the first procedure we compare directly the cut-off functions , see Figs. 4 and 4. They are defined as where is respectively for the numerical data, for the mean-field prediction, and for the simplest Padé approximant of the one-loop result, i.e. setting in (16). For , due to the large value of , the function with , , , and does not have the correct normalization. We chose to introduce two rescaling factors

(18)

in order to enforce the conditions (9) and (10). This procedure only changes the values of , and in a consistent manner, see Figs. 4 and 4. Note that even though only is plotted, the chosen value of changes the values of via the normalization conditions, hence must be discussed accordingly.

Figure 5: Random Field (). Blow up of the power-law region. The red solid curve is given by Eq.(19) , the black dashed line by Eq.(15), with , and .
Figure 6: Random Field (). Blow up of the tail region. The red solid curve is given by Eq.(19) , the black dashed line by Eq.(15), with , and .
Figure 5: Random Field (). Blow up of the power-law region. The red solid curve is given by Eq.(19) , the black dashed line by Eq.(15), with , and .

A second approach, shown in Figs. 6 and 6, consists in fitting the same numerical curves as in Figs. 4 and 4, with either (i) an exponential function (“exponential fit”) or (ii) the one-loop function (“fit one loop”), but using the numerically obtained exponent . The exponential fit reads

(19)

All coefficients are determined as a function of by the normalization conditions (9) and (10). Note that this exponential fit is mostly a guide to emphasize the sub-exponential tail apparent in the data. Similarly, for the one-loop fit we adopt the procedure described in the previous paragraph, with everywhere instead of the one-loop Padé value . We expect this fit to be less sensitive to the lack of precision in the one-loop estimate of for the large value of relevant here, and to better capture the tail region. This is indeed what is found, see Fig. 6. It confirms the sub-exponential tail exponent to a rather good precision. We stress that our procedure is not a fit using as fit parameters, but that all parameters are specified by the one-loop prediction.

Figure 7: Random Field (, for , ; for ). A fit with a power law gives access to the exponent . The comparison with Eq.(12) is discussed in the text.

We now turn to a 2-dimensional interface. The universal function for and RF disorder is plotted on Fig. 7. From a direct power-law fit, we find

(20)

This value has to be compared with the conjecture of Eq. (12). The roughness exponent at the depinning transition is known numerically as RossoHartmannKrauth2002 (), which gives . Although our value (20) of is compatible with the conjecture, the precision is insufficient to conclude on possible small deviations from the latter. The mean-field and one-loop predictions discussed above are plotted for comparison, using the simplest one-loop Padé approximant, i.e. , , . After the above described procedure (18) using the normalization conditions (9) and (10) this led to the values , and .

V Universal moment ratios

Figure 8: Numerical values of the ratios for bare random-field disorder (, ), and , . For a pole is expected for . For the divergence manifests itself through the data’s dependence on .
Figure 9: Random Field (, ). Moment ratios : comparison between numerics and analytic predictions
Figure 8: Numerical values of the ratios for bare random-field disorder (, ), and , . For a pole is expected for . For the divergence manifests itself through the data’s dependence on .
Figure 10: Random Field (, ). Moment ratios : blow up of the tail behavior. The mean field behavior is given by Eq.(22). The Padé by Eq(23) and the Padé is also plotted.
Figure 11: Random Field (, ). Moment ratios : blow up around the pole and comparison with Eq. (24).
Figure 10: Random Field (, ). Moment ratios : blow up of the tail behavior. The mean field behavior is given by Eq.(22). The Padé by Eq(23) and the Padé is also plotted.

Important universal quantities characterizing the avalanche statistics are the following universal ratios of avalanche-size moments:

(21)

Here can be non-integer. As shown in LeDoussalWiese2008c () all non-universal scales disappear in the ratios . Our numerical findings are summarized in Fig. 9. The pole expected at in the limit of infinite manifests itself in a non-convergence of the numerical data upon lowering . This is an independent method for calculating .

We now compare to the FRG calculation LeDoussalWiese2008c (). The function can be evaluated in an expansion. At the mean-field level ()

(22)

and a pole is found for .

The one-loop expansion leads to the following expression LeDoussalWiese2008c ():

(23)

where for RF. This expression corresponds to the Padé in the -expansion; we also use the Padé . The comparison with the data is shown on Fig. 9. For the large-moment region, a blow-up is shown on Fig. 11. The agreement of the data with the two one-loop Padé approximants, as compared to mean field, is quite striking.

Figure 12: The characteristic function for (left) and (right). Mean field (solid red/grey) from Eq. (29); one-loop from Eq. (30) (dashed black); and numerical results for (solid orange/green dots). The singularity in (29) and (30) for (indicated by a vertical dotted line) is smoothed out in the numerics, see figure 13 for details.
Figure 13: Left: Log-log plot of versus . For large , we expect from Eq. (15) that the analytic result (black, thick, dashed) obtained by integrating Eq. (28) numerically, has slope (blue dotted line is this asymptotics). The numerical results are for and (thick red) and (thin red). For large , , where is the largest avalanche encountered in the simulation, and the curve saturates (dotted red lines). The larger , the better the data. For (thick red) this ratio is 16, whereas for (thin red) it is 10. The vertical dashed green line indicates the location of the singularity in at the mean-field level. Right: Slope of the function left, i.e. the effective exponent . One sees that the effective exponent increases with increasing . Our data, which clearly have not converged in terms of , allow to estimate from the maximum slope.
Figure 12: The characteristic function for (left) and (right). Mean field (solid red/grey) from Eq. (29); one-loop from Eq. (30) (dashed black); and numerical results for (solid orange/green dots). The singularity in (29) and (30) for (indicated by a vertical dotted line) is smoothed out in the numerics, see figure 13 for details.

However, both Padés break down close to . We give another useful form for comparison to numerics. The idea is to isolate the simple pole which occurs in any dimension, as

(24)

Up to corrections LeDoussalWiese2008c ()

(25)
(26)

In Fig. 11 this formula is plotted setting . It shows that it works quite well, even close to the pole at .

Vi The characteristic function

It is useful to define a generating function of exponential moments, i.e. the characteristic function of the avalanche-size probability. Using the definitions (6) and (7), we define the normalized generating function

(27)

By construction, Since large negative probe small avalanches, it is expected to be universal for . In the universal range, its relation to is

(28)

It has been calculated in LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c () at the mean-field level:

(29)

At 1-loop order, it reads LeDoussalMiddletonWiese2008 (); LeDoussalWiese2008c ()

(30)

where, as above, . The comparison between theory and numerical data is presented on Fig. 13, both for , and . In these figures we have plotted (30), discarding the term and setting and respectively. The plots show that the simplest extrapolation of the 1-loop correction is extremely good in calculating the behavior even for large negative , as was already observed in the static case in LeDoussalMiddletonWiese2008 (). It would be interesting to compare for both cases numerically.

For large , is dominated by the largest avalanche . If , then the tail-exponent can in principle be extracted from the derivative of , i.e. in some window of before it eventually saturates to a constant at larger . Our data, which are plotted on Fig. 13, are not yet converged in terms of the ratio , but are sufficient to give the bound . One finally notes that although mean field works better for than for , the 1-loop corrections are necessary to account for the numerical data.

Vii Local avalanche-size distribution

Figure 14: Random Field (, , ). From the fit we get .
Figure 15: both at the tree-level (solid/red), and numerically (green/orange dots), for RF disorder, , , .
Figure 16: as defined in equation (38) and discussed below. For , .

In LeDoussalWiese2008c (), we have considered the following definition of the size of a local avalanche :

(31)

Here we also define:

(32)

Of particular interest is the cross-section with a co-dimension one hyper-plane i.e. , or more generally, with a co-dimension subspace. This cross-section has dimension . We have chosen the factor of in the definition of such that and both scale as . For we consider a point, i.e. , . Note that we always chose the factor of in the definition of (see above) such that and both scale as .

For a more convenient comparison with numerics, we adopt a slightly different normalization as in Ref. LeDoussalWiese2008c (), and chose to normalize using rather than . We estimate below the ratio which allows to go from one set of definitions to the other. Hence the (normalized) local avalanche-size distribution is expected to take the form

(33)

where the universal function in terms of the one defined in LeDoussalWiese2008c () and called there. By construction satisfies the normalizations (9) and (10). Similarly we define the generating function as

(34)

which reads in terms of the one defined in LeDoussalWiese2008c ().

At present time we have only three analytical results available to compare the numerical data on local avalanche-size distributions. First the conjecture put forward in LeDoussalWiese2008c () and which generalizes (12) reads:

(35)

where we recall that . Our numerical data for the point on a string (i.e. one monomer) is shown on Fig. 14 and we find . If we use the best present estimate we find . If we use the value of extracted from the scaling of , we find . While the values are roughly consistent, the precision on is crucial for a precise comparison. Inverting the conjecture (35), the measurement of leads to a conjectured . For a detailed discussion of the possible artifacts we refer to the discussion in Section IV.

The second result is the exact expression of and in mean field, i.e. for and . This involves a non-trivial summation of momentum-dependent tree diagrams using instanton calculus. It yields LeDoussalWiese2008c () that is given by the solution of

(36)

which vanishes at . This yields the series expansion . We have compared this mean-field prediction and the numerical results in , on figure 15. It is clear that loop corrections, yet to be computed, will play an important role, as was the case for bulk avalanches, see Fig. 13. The function , as defined here, is found to be LeDoussalWiese2008c () in mean field (i.e. at tree level)

(37)

It satisfies the normalizations (9) and (10), and is related to via .

Finally, the mean-field calculation LeDoussalWiese2008c () also gives:

(38)

Corrections at 1-loop order slightly decrease this ratio. The expansion predicts LeDoussalWieseToBePublished ()

(39)

with and here. Using the two Padé approximants gives the estimate which is consistent with the numerically observed value of for , .

Viii Conclusion

In this article, we have compared the numerically obtained avalanche-size statistics at the depinning transition with the recent predictions from the functional RG based on an expansion. The critical point of the depinning transition for an interface of internal dimensions and , driven quasi-statically in a random landscape in presence of an external quadratic well of curvature , is reached in the limit of . We have shown that the avalanche-size distribution takes the expected scaling form with the upper cutoff scale , involving a universal function in the rescaled variable . As we confirmed, it does not depend on whether the microscopic disorder is of random-field or random-bond type. We have computed numerically the function , its moments and its characteristic function and found in all cases good to excellent agreement with the predictions of the 1-loop FRG based on the extrapolation to and . We have also studied, for , the local avalanches, and there too, we found a rather satisfactory agreement with available analytical predictions. However, it remains an outstanding challenge to compute the local avalanche-size distribution within the FRG beyond mean-field.

Some fine points deserve discussion and further study. First we have not found any clear-cut signature that the conjecture for the avalanche exponent be violated at depinning. However, we can not rule out such a violation below a precision in . A better numerical determination of , comparable in precision to the one which exists for the roughness at would be crucial to confirm or invalidate the conjectured relation between and . Presently, the mass, i.e. the quadratic well, appears necessary for a proper definition of the steady state, but unfortunately, this hampers the attempts at a more precise determination of .

Second, there has been a recent proposal, in the case of the random-field Ising model LiuDahmen2006 (), that avalanche-size distributions for statics and depinning are described by the same universal functions. Although the physics underlying this hypothesis is not clear to us, one may still ask the question for the present model 333This hypothesis is proven incorrect in for the class of short-range correlated forces (random field class) where one finds on one hand the Sinai-model class in the statics LeDoussal2008 () and on the other, the Gumbel extremal statistics class at depinning LeDoussalWiese2008a (), with vastly different distributions in each class.. One may for instance compare our present results to the one in the statics in LeDoussalMiddletonWiese2008 (). Currently, our precision is not sufficient to conclude. For instance, in and for RF disorder, the conjecture (1) for gives for the statics and for depinning, which are difficult to distinguish numerically. We simply note that both statics and depinning data are in good agreement with extrapolations from the 1-loop FRG LeDoussalWiese2008c (); FedorenkoLeDoussalWieseInPrep (), but it remains to be analyzed at two loops. As noted previously, since the roughness exponents are different, if the conjecture holds both in the statics and driven dynamics, then the avalanche-size exponents, and presumably the associated distributions, cannot be the same. We leave these subtle questions for the future.

To conclude, it is highly satisfactory that the functional-RG field theory for the avalanche statistics passes all numerical tests. Other interesting observables can now be computed numerically, and studied on a more solid footing, such as the distribution of lateral sizes, or correlations between avalanches. These provide a motivation to further develop the theory. Finally we hope that our present work will motivate similar studies in experiments.

Acknowledgements.
This work was supported by ANR under program 05-BLAN-0099-01.

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