Exact results for a toy model exhibiting dynamic criticality
aa r X i v : . [ m a t h - ph ] D ec Exact results for a toy model exhibiting dynamiccriticality
David C. KasparMathematics DepartmentUniversity of CaliforniaBerkeley, CA 94720, [email protected] Muhittin MunganPhysics DepartmentBo˘gazi¸ci UniversityBebek 34342 Istanbul, [email protected] 23, 2018
Abstract
In this article we discuss an exactly solvable, one-dimensional, periodictoy charge density wave model introduced in [D.C. Kaspar, M. Mungan,EPL , 46002 (2013)]. In particular, driving the system with a uniformforce, we show that the depinning threshold configuration is an explicitfunction of the underlying disorder, as is the evolution from the negativethreshold to the positive threshold, the latter admitting a description interms of record sequences. This evolution is described by an avalanchealgorithm, which identifies a sequence of static configurations that arestable at successively stronger forcing, and is useful both for analysis andsimulation. We focus in particular on the behavior of the polarization P , which is related to the cumulative avalanche size, as a function of thethreshold force minus the current force ( F th − F ), as this has been thefocus of several prior numerical and analytical studies of CDW systems.The results presented are rigorous, with exceptions explicitly indicated,and show that the depinning transition in this model is indeed a dynamiccritical phenomenon. We consider an infinite chain of particles connected by springs, where each par-ticle is exposed to an external potential. The potentials are identical except forquenched random phase shifts. Such systems originally served as phenomeno-logical models for charge density waves, a quantum phenomenon observed incertain materials at low temperature [1], but are now considered model problemsin the study of the behavior of elastic manifolds in disordered media; see [2–4]for reviews.Under the influence of an external driving force, the particles move, perhapswithin a single well of the substrate, or from one well to another. If the external1orce is not too strong, the chain will, after some change in shape, come to rest;in this situation we say that the system is pinned , as there are positions for theparticles on the substrate which prevent the force from advancing it further. If,on the other hand, the force is very strong, no arrangement of the particles onthe substrate is sufficient to arrest its progress, and we have entered the sliding regime. The transition from one regime to the other occurs at a critical valueof the driving force, known as the threshold force F th . The behavior of thesystem near threshold, and in particular the transition from static to dynamicstates, has been a subject of interest in diverse areas, such as flux line latticesin type II superconductors [5], fluid invasion in porous media [6], propagationof cracks [7, 8], as well as models of friction and earthquakes [9].Fisher [10, 11] has argued that this depinning transition is an example of a dynamic critical phenomenon , a phase transition with the external force as thecontrol parameter. There is evidence to support this claim: • analysis [12, 13] of a different simplified model [14] for sliding particleswith random friction, showing the divergence of strains at the depinningthreshold, • functional renormalization group calculations [15–17], and • extensive numerical simulation in dimensions d = 1 , ,
3. [18–24]show or strongly suggest that certain properties of the system near thresholdexhibit scaling behavior. On the other hand, there are few rigorous results torely upon.In a short paper [25], the authors introduced a toy version of a CDW modelin one dimension which is exactly solvable: the threshold state is an explicitfunction of the underlying disorder, as is the externally forced evolution tothreshold through intermediate static configurations. This permits a preciseexamination of certain observables, particularly the cumulative avalanche size ,which is related to the CDW polarization , and here we find the tell-tale signs ofa critical phenomenon. In this article we provide the proofs and further detailsof the results stated in [25].The paper is organized as follows. In Section 2 we describe the Fukuyama-Lee-Rice model for CDWs, and the toy model approximation that results fromtruncating the range of interactions. We introduce also the observables westudy as the configurations in these systems are driven to threshold. Next, inSection 3 we formalize the process of evolving a given configuration to threshold,through a sequence of static configurations, as the avalanche algorithm . Anumber of associated results hold for both the toy model and the untruncatedversion. Section 4 presents additional observations for the toy model, which takeparticular advantage of the explicit description of the threshold state availablein this case. Both Sections 3 and 4 concern statements which hold almostsurely with respect to the underlying disorder; in Section 5 we turn to statistics.Remarks regarding numerics are found in Section 6. To develop the precedingmaterial free from distraction, we defer all proofs to Section 7. Lastly, in Section2 we discuss our work and its context in the existing literature, and indicateremaining questions for future work.
The Fukuyama-Lee-Rice [26,27] description of CDWs is analogous to a bi-infinitechain of particles connected by springs, where each particle is subject to arandomly shifted potential. We assume also the presence of an external forceacting uniformly on all the particles. A formal Hamiltonian for such a systemis H ( { y i } ) = X i ∈ Z
12 ( y i − y i − − µ ) + V ( y i − α i ) − F y i . (2.1)Each particle i is constrained to move in only one direction; we call its locationalong this line y i . We have assumed the springs are Hookean with equilibriumlength µ , and normalized their common stiffness. The potential V is 1-periodic,and each particle sees a different random translate of it. F is the driving forceapplied uniformly to all the particles. Particular choices are suitable for derivingexact formulas: • Let α i be i.i.d. uniform ( − , + ). As V is 1-periodic, we may as wellregard our random shifts as elements of the circle, where the Lebesguemeasure is a natural choice. • As in [15, 28], we select the potential V as V ( x ) = λ x − J x K ) , (2.2) J x K denoting the integer nearest to x . The parameter λ > V and the springs.Figure 1 illustrates the situation.It is possible to study the dynamics of such a system, under a time-varyingforce, with a system of ODEs for inertialess particles under relaxational dynam-ics: other authors such as [18–20,23,27] have pursued this approach. Instead wewill assume that the time scale at which the external force is changing is muchlarger than that associated with the relaxation of the particles, and thereforeconsider the approach to threshold through intermediate static configurations.Lemma 3.4 and its analogue in the dynamic case, the no passing rule of [23],indicate some manner of equivalence between these approaches.Static configurations are those for which ∂ y i H = 0 for all i : − ∆ y i + V ′ ( y i − α i ) − F = 0 . (2.3)Here ∆ denotes the discrete Laplace operator on sequences given by ∆ y i = y i − − y i + y i +1 . When using the potential (2.2) it is convenient to introduce3 V ( y ) (a) y i F (b)Figure 1: (color online) Illustration (a) shows the shape of the potential V ( x ),while (b) visualizes a portion of the bi-infinite chain of particles. In (b) theparticles are marked with blue dots and the “springs” connecting them aredashed red lines. The vertical black lines show the sequence of potential wellsseen by particle i , with horizontal markings to indicate the cusps of V . Anarrow indicates the direction of the external force F exerted on the particles.the notation m i ≡ J y i − α i K ∈ Z (2.4a)˜ y i ≡ y i − α i − m i ∈ (cid:0) − , + (cid:3) ; (2.4b)we refer to these as the well number and well coordinate of y i , the former indi-cating which parabolic well contains the particle and the latter the displacementof the particle from the center of its well. Then (2.3) can be re-expressed as( λ − ∆) y = λ ( m + α + F/λ ) . (2.5)As in [28], we may treat m and α as given and solve this linear equation for y .It is important to note, however, that the nonlinearity of this system has not4isappeared, but rather it becomes a consistency condition: after computing y from m , we must have m i = J y i − α i K for all i .Elementary techniques for linear recurrences applied to (2.5) give a formulafor y : y i = 1 − η η X j ∈ Z η | i − j | ( m j + α j ) + Fλ , (2.6)where η = 22 + λ + √ λ + 4 λ ∈ (0 , . (2.7)This is the unique choice for which y i does not grow geometrically as | i | → ∞ even if m + α is bounded. Noting that λ ˜ y = ∆ y + F from (2.5), it follows alsothat ˜ y i = η − η X j ∈ Z η | i − j | (∆ m j + ∆ α j ) + Fλ . (2.8)Momentarily ignoring the relationship between y and m , observe that increas-ing or decreasing the driving force F affects the configuration by rigid transla-tion. This, and the explicit formulas (2.6) and (2.8), are the advantages of theparabolic potential.Taking a configuration (with F = 0, for example) and increasing F causesthe chain of particles to rigidly translate until for some i we have ˜ y i = ; anyfurther increase causes this particle to topple into the next well. See Figure 2for illustration. For instance, if a jump occurs at site j , the resulting change inwell coordinates is ˜ y i → ˜ y i − δ ij + 1 − η η η | i − j | (2.9)provided that ˜ y i < for all i after the change; otherwise, other particles will bepulled forward into their next wells. This process may terminate, resulting in anew static configuration, or continue forever, in which case we understand theconfiguration is no longer pinned and has entered the sliding regime.For this model we are interested in answering the following questions: • At what F does the system depin and enter the sliding regime? We callthis F the threshold force and denote it F th . • What is the shape of the configuration just before it begins to slide? As | y i − m i | < i by definition, the well numbers we observe just beforethe threshold, m + , sufficiently describe the large-scale shape. • How do various observables behave in terms of F th − F ? We are particu-larly interested in the polarization , which is the spatial average (i.e. averageover i ) of the change in m i as we evolve from some initial configuration tothe first configuration we encounter that is stable at the current force F .In subsequent sections we present theoretical and numerical results for a finite version of this system with periodic boundary conditions. For a system with L particles, we take the well numbers m i and the disorder α i to be L -periodic5a)(b)Figure 2: The configuration is (a) rigidly translated until a particle reaches theedge of its well (red) and then (b) this particle jumps into the next well, pullingall the other particles forward by an amount that decays geometrically movingaway from the site that jumped.sequences, the latter still i.i.d. within a single period. Our most detailed resultsare for an approximation we call the toy model [25]. For a strong potential, λ is large and η is very small, and we have˜ y i = η (∆ m i + ∆ α i ) + F/λ + O ( η ) . (2.10)Dropping the O ( η ) portion reduces the range of direct interactions to nearestneighbors only. In this case we can answer very explicitly all the questions posedabove. Our basic tool for both simulation and the derivation of rigorous results is the avalanche algorithm . This takes as input a static configuration, and produces anew static configuration which is stable at higher force, if possible, in a mannerintended to mimic the result of increasing the force and finding the long timelimiting arrangement of the particles with an inertialess dynamics. For the L -periodic chain in both the toy model and the model with long-range interaction,we describe this procedure in terms of the well numbers m and well coordinates˜ y from (2.4a) and (2.4b). 6 lgorithm 3.1 (avalanche with force) . Given a valid configuration m in theenvironment specified by α and F , we produce a new configuration m ∗ valid ata new F ∗ ≥ F :(A1) Start with the current configuration: let m ∗ = m (and correspondingly˜ y ∗ = ˜ y ) initially.(A2) Record the maximum well coordinate ˜ y max = max i ˜ y i .(A3) Increase the force to F ∗ = F + λ ( − ˜ y max ), and correspondingly adjustthe well coordinates ˜ y ∗ i → ˜ y ∗ i + ( − ˜ y max ), bringing exactly one particle(in each period) to the cusp of the next well.(A4) Let j = arg max i ˜ y ∗ i and jump particle j (and its periodic equivalents) byincrementing m ∗ j → m ∗ j + 1 and suitably adjusting ˜ y ∗ : for the full model,˜ y ∗ i → ˜ y ∗ i + − η η + 1 − η η η L − η L for i = j − η η η | i − j | + η L −| i − j | − η L for j < i < j + L (3.1)and for the toy model,˜ y ∗ i → ˜ y ∗ i + ( +1 η for i = j ± − η for i = j, (3.2)and extending these periodically.(A5) If ˜ y ∗ i > / i , goto (A4). Remark.
The formula (3.1) for updating ˜ y has been adapted from (2.8) torespect the periodicity. For simulation purposes we might use (2.8) unaltered,summing only over nearest periodic representatives, at the cost of an O ( η L )error.For this algorithm to be well-defined, we must verify that it does, in fact,terminate. The following result indicates that it does, and gives the maximumnumber of jumps (A4) we might expect. It also establishes a useful propertywhich will allow us to give a centered version of the algorithm, which is betternumerically, requiring fewer floating point operations, and better theoretically,allowing us to recast the evolution in terms of a variational problem. Proposition 3.2.
The avalanche algorithm 3.1 has the following properties:(i) It terminates after finitely many steps.(ii) All particles jump at most once: m ∗ ≤ m + , the inequality holdingcomponentwise. iii) If F ≥ and η < / , then for all i the resulting configuration has ˜ y ∗ i > −
12 + F ∗ − Fλ . (3.3)Property (i) is immediate from (ii), which itself follows from a considerationof (3.1) or (3.2): a particle i which jumps once, decreasing ˜ y i , will not seesufficient increase in ˜ y i to exceed its original height, even if all the other particlesjump. Property (iii) tells us that the configuration m ∗ at force F ∗ produced bythe algorithm remains a valid configuration—that is, has all its well coordinatesin ( − , + ]—at the original force F . This illustrates that the models underconsideration exhibit both reversible and irreversible behavior : increasing theforce from 0 to some F > , F ] moving by rigid translation only, i.e. reversibly. Italso allows us to write a simpler algorithm which will produce the exact same sequence of configurations. Algorithm 3.3 (zero-force avalanche) . Given a configuration m in an envi-ronment specified by α with F = 0, produce a new configuration m ∗ valid at F ∗ = 0:(ZFA1) Let m ∗ = m .(ZFA2) Record ˜ y max = max i ˜ y i .(ZFA3) Let j = arg max i ˜ y ∗ i and jump particle j (and its periodic equivalents) byincrementing m ∗ j → m ∗ j + 1 and correspondingly adjusting ˜ y ∗ as in (3.1)or (3.2).(ZFA4) If ˜ y ∗ i > ˜ y max for any i , goto (ZFA3).For brevity we refer to this algorithm as the ZFA . Note that the result hasmax i ˜ y ∗ i = ˜ y ∗ max ≤ ˜ y max .Middleton’s no passing rule [23] is a monotonicity property of the inertialessODE system used to study CDWs from a dynamic perspective: if y ( t ) and y ( t )are two solutions to ˙ y = −∇H ( y ) where y ( t ) ≤ y ( t ), then y ( t ) ≤ y ( t )for t ≥ t . Monotonicity results are an essential tool for studying arrangementsof chains of particles, even in a purely static setting. Consider, for example,the Aubry-Mather treatment of the similar Frenkel-Kontorova model, explainedvery nicely by Bangert [29]. That the ZFA has such a property is necessary forour subsequent observations. Lemma 3.4 (ZFA noncrossing) . Let m ≤ m be two configurations for eitherthe full or toy model sharing the same environment α , and let m ∗ and m ∗ bethe results of applying the ZFA to each of these. More precisely, the well numbers produced will be exactly the same, and the well coordi-nates will differ only by an overall translation applied uniformly to all particles. i) If max i ˜ y i > max i ˜ y i , then m ∗ ≤ m .(ii) If max i ˜ y i = max i ˜ y i and m j < m j for j = arg max i ˜ y i , then m ∗ ≤ m .(iii) If max i ˜ y ≥ max i ˜ y i , then m ∗ ≤ m ∗ . In each case above, the stated conditions give a bound on the well coordinatesof any particle i for which m i = m i , which prevents particle i from jumpingin cases (i) and (ii), or shows that particle i jumps for configuration 1 only ifit jumps for configuration 2. The argument is very much the same as for thedynamic version [23].We now define the threshold states for the full and toy models. Consideringthe above noncrossing result, the threshold configuration should be that whichminimizes max i ˜ y i : another configuration could not depin without first crossingthis one. Definition 3.5.
In either the full model or the toy model, for a given environ-ment α , a threshold configuration is specified by well numbers m + achievingmin m max i ˜ y i (3.4)where ˜ y is the vector of well coordinates corresponding to m at F = 0. The threshold force is F th = λ (cid:18) − min m max i ˜ y i (cid:19) . (3.5)Note that F th is exactly the force required to bring one particle in the thresholdconfiguration to the upper edge of its well. Here and in the sequel, we computewell coordinates from well numbers at F = 0. Remark.
With standard Frenkel-Kontorova, one is interested in configurationswhich minimize energy, which consists (in the case of Hookean springs) of an ℓ -difference of y and its translate by one, and the terms coming from thesubstrate potential. Here, when considering a similar system in the presence ofan increasing driving force, the relevant functional is of ℓ ∞ -type.Our next result illustrates the utility of the ZFA as we try to understandthreshold behavior. Proposition 3.6.
For both the full model and the toy model:(i) The threshold configuration m + exists and is almost surely unique, up totranslating all components of m + by the same integer.(ii) Starting from m = , the ZFA finds m + in finitely many steps.(iii) The ZFA applied to m + produces m + + , and this property is unique tothe family m + + Z . i | ∆ m i | , allowing us to exclude all but a finite set of m (modulouniform translation by integers). Uniqueness is also relatively routine, afterusing the noncrossing property of the ZFA to reduce possible nonuniqueness toconfigurations which are ordered and have well numbers differing by at mostone. Noncrossing gives (ii), and the uniqueness, together with the fact that theZFA can never increase max i ˜ y i , implies (iii).We thus have a tool, the ZFA, for both the full and toy models, whichproduces the threshold configuration that precedes the depinning transition. Itachieves this by way of a sequence of physically meaningful intermediate states,according to an algorithm which is straightforward to implement and apply togenerate numerical results. In the next section we specialize to the toy model,where more can be said. In the case of the toy model we find it convenient to introduce rescaled wellcoordinates z defined by ηz i = ˜ y i . (4.1)As in the previous section, we fix the external force F = 0. In this case, a jumpat site j as in step (iii) of Algorithm 3.3 results in m j → m j + 1 , z j → z j − , z j ± → z j ± + 1 . (4.2)Here we find a strong similarity between the toy model and sandpile models(see [30] for an introduction), as already noted by other authors working onsimilar CDW systems [20, 24, 31]. Indeed, for one-dimensional sandpile models,the change to z in (4.2) is precisely the result of toppling at site j . The existingliterature on sandpiles is extensive; see [32] for a survey, and note that modelswith continuous heights have been considered previously [33]. However, theauthors are unable to find an exact match for the toy model in prior work. Asnoted in [25], the toy model has periodic boundary, conserves the sum of z ,evolves deterministically, changes by integers only, and preserves the fractionalpart of the ∆ α . We discuss this connection further in Section 5.For now, the similarity between the two is a sign to expect that the toymodel will permit exact results: the set of recurrent states of a standard one-dimensional sandpile is rather trivial, and one might hope that the toy model’spersistent disorder does not introduce so much complexity that things becomeintractable. The primary result of this section confirms this: the solution of theoptimization problem posed in Definition 3.5 can be expressed explicitly. Theorem 4.1.
Let S = P L − i =0 J ∆ α i K . The a.s. unique threshold configurationfor the toy model m + satisfies ∆ m + i = − J ∆ α i K + J i − δ ik + (4.3) where J is an integer vector selected as follows: Case S ≥ J i = 1 for the S + 1 positions i which have smallest ∆ α i − J ∆ α i K and J i = 0 otherwise. • Case
S < J i = − for the | S | − positions i which have largest ∆ α i − J ∆ α i K and J i = 0 otherwise.and k + is an index defined by k + = L − X i =0 i ( − J ∆ α i K + J i ) (mod L ) . (4.4)The proof is given in Section 7, and, due to Proposition 3.6, amounts tochecking that m + is mapped to m + + by the ZFA. To explore the consequencesof this explicit description, we first require some notation. Let ǫ i = ∆ m i + J ∆ α i K , (4.5)and refer to those sites where ǫ i = 0 as defects with charge ǫ i . Write ω i = ∆ α i − J ∆ α i K (4.6)for the fractional part of ∆ α i , and let σ be the permutation of { , , . . . , L − } which orders ω : ω σ (0) < ω σ (1) < · · · < ω σ ( L − . (4.7)Using this terminology, Theorem 4.1 gives the threshold force explicitly. Corollary 4.2.
For the toy model, the maximum z +max of the rescaled wellcoordinates (see (4.1) ) of the threshold configuration is z +max = ω σ ( S ) + 1 if S ≥ and k + = σ ( S ) ω σ ( S − + 1 if S > and k + = σ ( S ) ω σ ( L − if S = 0 and k + = σ (0) ω σ ( L −| S | ) if S < and k + = σ ( L − | S | ) ω σ ( L −| S |− if S < and k + = σ ( L − | S | ) , (4.8) and the corresponding threshold force F th is F th = λ (cid:18) − ηz +max (cid:19) . (4.9) Remark.
As we will see in Section 5, the cases k + ∈ { σ ( S ) , σ ( L − | S | ) } haveprobability tending to 0 as the system size L → ∞ .We wish to understand not only the threshold configuration but the behav-ior of the system as we approach it. The noncrossing property Lemma 3.4 ofthe ZFA implies that we may take any valid configuration and, by repeatedapplication of this algorithm, arrive at the threshold state. During this process,we track certain quantities associated with the system’s evolution.11f particular interest is the observable known as polarization, as this hasbeen the subject of several previous studies in CDW and related models [15,20,24]. Given an initial state corresponding to some well numbers m , applyingthe ZFA produces a sequence of configurations (essentially) terminating with m + . Suppose that we record these configurations, calling them m ( τ ) for τ insome index set T . Then the polarization is the function of τ given by P ( τ ) ≡ L L − X j =0 ( m j ( τ ) − m j ) . (4.10)We write also Σ( τ ) = LP ( τ ), and call Σ( τ ) the cumulative avalanche size . Ineither case, the quantity under consideration is the total number of particlejumps which have occurred in the process of evolving from the initial state m to the current state m ( τ ).Among all possible initial conditions m , two seem particularly natural froma macroscopic perspective: we might begin with flat well numbers, m i = 0 forall i , or we might take the negative threshold configuration, m = m − , definedprecisely below. In the flat case we have only statistics for the complete evo-lution without intermediate configurations, which we discuss in Section 5. Forthe threshold-to-threshold evolution, on the other hand, there is a nice interpre-tation, in terms of record sequences, for each step of the evolution, which wedevelop in the remainder of this section.For both the toy model and the full model, given a realization α , write m + for a threshold configuration as previously defined, and call it a (+) -thresholdconfiguration . Define also a ( − ) -threshold configuration m − , which achievesmax m min i ˜ y i (4.11)for ˜ y i the well coordinates corresponding to m . (Note that this can be obtainedfrom the (+)-threshold configuration with α replaced with − α .) We can adapt(4.3) to produce the negative threshold configuration of the toy model, whichmaximizes min i z i . Define J − and k − as follows:(i) Case
S > . J − i = 1 for the S − i which have smallest ω i , J − i = 0 otherwise;(ii) Case S ≤ . J − i = − | S | + 1 positions i which have largest ω i , J − i = 0 otherwise;and k − is given in terms of J − by analogy with (4.4).In the toy model, several successive applications of the ZFA may have initialjumps occurring at the same site. This behavior will be especially prevalentfor the threshold-to-threshold evolution. It will be useful both intuitively andtechnically to view these transitions in aggregate. For a given non-threshold configuration ( m , z ) with i = arg max k z k , let i L and i R be the indices of sitesclosest to i on the left and right, respectively, for which z i L + 1 ≤ z i and z i R + 1 ≤ z i . (4.12)12efine sets of indices W , . . . , W ℓ for ℓ = ( i − i L ) ∧ ( i R − i ), by W k = [ i L + k, i R − k ] . (4.13) Proposition 4.3.
The first ℓ iterations of the ZFA applied to ( m , z ) as abovecause jumps at sites with indices in W , . . . , W ℓ , respectively. We call this se-quence an avalanche , the individual iterations avalanche waves , and i L , i R the left and right extents of the avalanche. (The wave terminology has been bor-rowed from sandpile models [34].) Throughout this process, site i remains thelocation of the maximum, and the original z i the maximum value, at least untilafter the ℓ th iteration. It follows that(i) The total number of jumps in the avalanche is ( i − i L )( i R − i ) .(ii) The resulting changes in the configuration ( m , z ) are: z i L → z i L + 1 (4.14a) z i R → z i R + 1 (4.14b) z i → z i − z i L + i R − i → z i L + i R − i − and m j → m j + ( j − i L ) + − ( j − i ) + − ( j − i − i R + i L ) + + ( j − i R ) + . (4.14e) In the case where i − i L = i R − i , the transition at i is z i → z i − . Note the change δ m in m shown in (4.14e) is trapezoidal. Figure 3 displaysthe changes resulting from several avalanches. To illustrate the threshold-to-threshold evolution for the toy model, it is convenient to introduce ζ i = ω i + J − i (4.15)and the permutation π that orders ζ : ζ π (0) < ζ π (1) < · · · < ζ π ( L − . (4.16)Note ζ π ( L − − ζ π (0) <
1. The ( ± )-threshold configurations have z − i = ζ i + δ ik − (4.17) z + i = ζ i + δ iπ (0) + δ iπ (1) − δ ik + , (4.18)and, using the divisibility condition (4.4), k ± are related by k + = π (0) + π (1) − k − . (4.19) We use the notation a ∧ b = min( a, b ) and a ∨ b = max( a, b ). Likewise, x + = 0 ∨ x . Here and in the the following addition and subtraction of indices are mod L . m under the ZFA starting from the negative thresholdconfiguration (blue). The sequence of intermediate configurations reached bytriggering of avalanches is shown in red. The topmost configuration is the posi-tive threshold configuration. The inset displays the trapezoidal change resultingfrom one of the avalanches.Observe that the ranks π − of the ζ suffice to determine an avalanche’sinitial site and extents. We represent a given configuration z j by displayingthe rank π − ( j ) of ζ j and using over- or underlines to indicate additions by ± s ↔ z π ( s ) = ζ π ( s ) + 1 ,s ↔ z π ( s ) = ζ π ( s ) − . (4.20)As in [25], an example clarifies things. Suppose that z − has the rank repre-sentation . . . . . . so that k − = π (15). The extents of the first avalanche are k − − i L = 2 , i R − k − =3 and after the sites bracketed below have jumped, the resulting configurationis . . . . . . . In the second wave, k − and k − + 1 jump again, yielding . . . . . . , lower records [35, 36]: given a sequence of values X , X , . . . ,we say that X i is a lower record if X i = min { X j : j ≤ i } . Using (4.12) andProposition 4.3 we see that avalanches are determined by the locations of thelower records of the sequences J L = ζ k − , ζ k − − , ζ k − − , . . . , ζ π L , (4.21)and J R = ζ k − , ζ k − +1 , ζ k − +2 , . . . , ζ π R , (4.22)where { π L , π R } = { π (0) , π (1) } are the termination sites. The evolution fromnegative to positive threshold terminates when the avalanches reach π L and π R .We are most interested in the dependence of the polarization on F th − F , thedifference between the current force and that at (+)-threshold. For the zero-force description, the quantity that serves this purpose is X ≡ z max − z +max , themaximum height of the current configuration minus that of the (+)-thresholdconfiguration. We parametrize the configurations we see in the threshold-to-threshold evolution by a nonnegative real quantity x : m ( x ) is the first config-uration we see for which X ≤ x . Note that this has the effect of skipping overthe results of the individual avalanche waves, because only complete avalanchesgive a strict decrease in X .By shifting indices, we can make k − = 0; let j L ( x ) and j R ( x ) be the (non-inclusive) left and right extents of the interval of sites which have jumped inorder to achieve X ≤ x . We select − L + j R ( x ) < j L ( x ) ≤ ≤ j R ( x ) < j L ( x ) + L. (4.23)Note that j L ( x ) and j R ( x ) are indices of the lower records from the sequences(4.21) and (4.22), respectively. In the threshold-to-threshold evolution, j L ( x )and j R ( x ) are sufficient to characterize the shape of m ( x ) − m , because thisremains trapezoidal. This follows because • the result of any complete avalanche is a trapezoidal change, and • for the threshold-to-threshold evolution, starting with an overall trape-zoidal change, the next avalanche is initiated at one of its convex corners,and terminates on one side at one of the concave corners.Then the corresponding cumulative avalanche size Σ( x ) and polarization P ( x )are Σ( x ) = − j L ( x ) j R ( x ) (4.24) P ( x ) = Σ( x ) L . (4.25)To understand the threshold-to-threshold polarization as a function of F th − F amounts to understanding the statistics of the pair j L , j R . This and otherprobabilistic questions are addressed in the next section.15 Statistical results
We begin by characterizing the variates ω i = ∆ α i − J ∆ α i K introduced previously,as the ( ± )-threshold configurations are explicit functions of these. The followingproposition is not interesting itself, but gives some indication how the choice wehave made for the disorder enables the subsequent results. Proposition 5.1.
The variates ω i = ∆ α i − J ∆ α i K , i = 0 , . . . , L − , havethe joint distribution that results from taking i.i.d. uniform ( − , + ) variatesand conditioning them to sum to an integer; by this we mean ω is distributedaccording to the (normalized) surface measure on the intersection of the cube ( − , + ) L with the family of planes x + x + · · · + x L − ∈ Z . Using the above it is easy to check that the one-dimensional marginals areuniform ( − , + ), and while { ω i } L − i =0 are dependent, removing just one of theseis enough to restore independence. We apply the central limit theorem for L − . Corollary 5.2.
The sum S = P L − i =0 J ∆ α i K , and hence the number of topologicaldefects, behave as follows as L → ∞ .(i) As L → ∞ , L − / S = L − / L − X i =0 J ∆ α i K (5.1) converges in distribution to a normal random variable with mean 0 andvariance / .(ii) The typical number of topological defects (sites where ǫ i = ∆ m i + J ∆ α i K =0 ) scales like L / .We observe also numerically that as L → ∞ , E F th → λ (1 − η ) and L / ( F th − E F th ) converges in distribution to a Gaussian with mean 0 and variance (12 L ) − The rescaled well coordinates z + at threshold are obtained by the modifica-tion of ω described in (4.3) and (4.4). This modification does not preserve allthe properties of ω , but a particularly important one is left intact. Theorem 5.3.
The components z + i of the vector z + of centered, rescaled well-coordinates at threshold are exchangeable. This leads quickly to a nice macroscopic description of the threshold con-figurations as L → ∞ . First, some physical motivation: the strains , whichare the magnitudes of the forces exerted by springs connecting the particles,are expected to diverge at threshold [12, 13] in CDW systems. This is possiblebecause we have assumed that the interaction between the particles can sur-vive any stress applied to it. This is of course unphysical and one expects thatbeyond a certain strain, plastic effects become dominant. In the case of CDWsystems, this plasticity gives rise to phase slips: the springs yield once the strain16eaches a certain value. If we intend to use the toy model to better understandsuch behavior, we need to understand how the strains build up as a functionof the external force. At present, we can at least characterize the strains atthreshold in a precise way.Write s i = m i +1 − m i , i = 0 , . . . , L −
1, for the strains in the configurationindicated by m . Also let s ( L ) ( t ) ≡ (12 /L ) / s ⌊ Lt ⌋ (0 ≤ t ≤
1) (5.2)be the c`adl`ag process obtained from s after central limit rescaling. A wellknown limit theorem for exchangeable variates (found for instance in [37]) givesthe distributional limit of the processes s ( L ) . Corollary 5.4.
With m = m + and the corresponding threshold strains s , as L → ∞ the processes s ( L ) converge distributionally in the Skorokhod space D ([0 , (equipped with the J -topology) to a periodic Brownian motion withzero integral: B ( t ) ≡ B ( t ) − Z B ( r ) dr (0 ≤ t ≤ , (5.3) where B ( t ) is a standard Brownian bridge. The process B is Gaussian withzero mean, stationary under periodic translations of the interval [0 , , with co-variance given by E B (0) B ( t ) = 112 (1 − t + 6 t ) (0 ≤ t ≤ . (5.4)Simulations of full CDW systems [15, 24] suggest that the total threshold-to-threshold polarization scales like P ∼ L / . The scaling limit of Corollary5.4 allows us to deduce this scaling for the total polarization from flat initialcondition to threshold. We compute P = 1 L L − X i =0 m + i = Z m + ⌊ Lt ⌋ dt = Z ⌊ Lt ⌋ X i =0 s i − min ≤ r ≤ ⌊ Lr ⌋ X i =0 s i dt = L Z (cid:26)Z t s ⌊ Lu ⌋ du − min ≤ r ≤ Z r s ⌊ Lu ⌋ du (cid:27) dt = L / (cid:26)Z Z t L − / s ⌊ Lu ⌋ du dt − min ≤ r ≤ Z r L − / s ⌊ Lu ⌋ du (cid:27) = L / √ (cid:26)Z s ( L ) ( t )(1 − t ) dt − min ≤ r ≤ Z r s ( L ) ( t ) dt (cid:27) The functional on D ([0 , ψ ( t ) Z ψ ( t )(1 − t ) dt − min ≤ r ≤ Z r ψ ( t ) dt (5.5)is continuous, so this yields a distributional limit for L − / P .17he distributional limit for p /L P can be re-expressed in terms of Brow-nian bridge: Z B ( t )(1 − t ) dt − min ≤ r ≤ Z r B ( t ) dt = max ≤ r ≤ Z B ( t )( − ( t − r ) − (0 ,r ) ( t )) dt. (5.6)Writing φ ( t ) = − t for 0 ≤ t ≤
1, and extending so that φ is 1-periodic, thedesired distribution is that ofmax ≤ r ≤ Z B ( t ) φ ( t − r ) dt = max ≤ r ≤ Z B ( t + r ) φ ( t ) dt, (5.7)extending B ( t ) to be 1-periodic. Noting that B ( · + r ) − B ( r ) has the samedistribution as B ( · ), and that φ ( t ) is orthogonal to constant functions, we findthat G ( r ) = Z B ( t ) φ ( t − r ) dt (5.8)is a mean zero, stationary Gaussian process. A straightforward calculation gives E G (0) G ( r ) = 1720 (1 − r + 60 r − r ) (0 ≤ r ≤ . (5.9)In particular, E ( G ( r ) − G (0)) ∼ r as r →
0, and a result of Weber [38] appliesto show there exists a constant c > c − t √ t √ ≤ P (cid:26) max ≤ r ≤ G ( r ) > t (cid:27) ≤ ct √ t √ t ≥
0. Here Ψ( x ) is the probability that a standard normal randomvariable exceeds x . It follows that the distributional limit of L − / P has sub-Gaussian tail. We are unable to describe the distribution more precisely, and ingeneral distributions of maxima of Gaussian processes are known explicitly inonly a handful of cases [39]. See Figure 4 for simulation results.For the threshold-to-threshold polarization in the toy model, our descriptionis considerably more detailed: instead of a single quantity P , we have a function P ( x ) defined in (4.25) with a parameter x indicating how close we are to thethreshold ( x = 0). Interestingly, the threshold-to-threshold polarization P (0) ∼ L , not L / .Relating the following proposition to the genuine P ( x ) requires an approxi-mation which remains, at the moment, unjustified , but the result seems reason-able and matches very well our simulations. Proposition 5.5.
Approximate J L and J R from (4.21) and (4.22) with i.i.d.uniform ( − , + ) variates sharing their first elements. Writing x = u/L , weobtain the finite-size scaling function Φ( u ) for the cumulative avalanche size: Φ( u ) ≡ lim L →∞ L − E [Σ( u/L )] = 6 − u + u − e − u − ue − u u . (5.11)18igure 4: Simulated distribution for the flat-to-threshold polarization, rescaledby L − / , for various L . The distributions were obtained from 10 randomrealizations for each size. This is the result of averaging the distributional limit ς ( u ) ≡ lim L →∞ Σ( u/L ) /L ,which has density p u ( s ) = P ( ς ( u ) ∈ ds ) /ds given by p u ( s ) = Z √ s dz e − zu u (1 − z ) + 2 u (1 − z ) ( z − s ) / , (5.12) with support on the interval [0 , ] . Some remarks are needed to interpret this result. First note that there is nosingularity in (5.11): writing series for the exponentials,Φ( u ) = 112 − u
30 + u − u
105 + O ( u ) (5.13)for 0 < u ≪
1. For u ≫ u ) ∼ u − . (5.14)Noting the definition (5.11), this shows that Σ (and not P , [25]) exhibits finitesize scaling behavior: i.e. the graphs of L − E [Σ] vs. u = XL for various L asymptotically collapse to the graph of a the scaling function Φ( u ) and moreover,in the scaling regime u ≫
1, the dependence on L drops out. This is indeedconfirmed by the results of numerical simulations shown in Figure 5. The finitesize scaling behavior implies that the correlation length ξ scales as ξ = X − . (5.15)19n terms of the underlying record process we can motivate this as follows. Givena current record X , the next record will occur on average after 1 /X sites. Sinceall sites within this range are forced to jump once the current record site initiatesthe next avalanche, this defines the correlation length ξ ∼ X − ν , with exponent ν = 1. The crossover to the saturated regime occurs when ξ is comparable to L , namely u = XL ∼ L/ξ ∼
1. From (4.24), the cumulative avalanche size isthe product of the left and right extents of sites which have jumped, and thusscales as X − . This exponent is traditionally denoted as − γ + 1 [11,24], so that γ = 3. The crossover behavior is clearly seen in Figure 5.Figure 5: Numerical results for the expected cumulative jump size Σ( X ) vs.the reduced and rescaled force, u = XL , in the evolution from the negative topositive threshold configuration. Symbol colors refer to different system sizes L , as indicated in the legend, with accompanying numbers of realizations inparentheses. The blue dashed line indicates a power-law with exponent − E (Σ) L − = 1 /
12. The solid lineis the theoretical finite-size scaling function, Φ( u ), (5.11).Two observations can be made about the distribution of ς ( u ). We first iden-tify a rescaling demonstrating its scale-free behavior within the scaling regime,and then simplify (5.12) in the case u = 0. In each case, the results take familiarforms. Making a change of variable t = uz − u √ s, the integral (5.12) becomes p u ( s ) = e − u √ s Z u (1 − √ s )0 dt e − t u − t − u √ s ) + 2( u − t − u √ s ) p t ( t + 4 u √ s ) . (5.16)20igure 6: Numerical cumulative avalanche size distribution for various L and u . For large u , the distributions collapse when avalanche sizes are scaled as a = u s . The solid line is (5.12). Symbol colors refer to different L , as indicatedin the legend and the numbers of realizations are shown in parentheses. Symbolshapes refer to the different values of u chosen.Scaling ς ( u ) such that lim u →∞ u ς ( u ) ≡ a , (5.17)the right endpoint of the interval of integration in (5.16) tends to ∞ , and thenumerator of the fraction in the integrand is 2 u to leading order. By dominatedconvergence as u → ∞ , the density p ( a ) = P ( a ∈ da ) of the rescaled avalanchesize a is p ( a ) = 2 e − √ a Z ∞ dt e − t p t ( t + 4 √ a ) = 2K o (2 √ a ) , (5.18)where K o is the modified Bessel Function, which decays at large values of itsargument as e − √ a / (2 √ a ) / . For the u -values shown in Figure 6, the asymp-totic form (5.18) is indistinguishable from the exact result (5.12), explaining thecollapse of the data. The form of the scaling variable a can be understood bynoting that a = u s = X Σ = Σ /ξ ; thus the avalanche sizes are measured inunits of ξ .Next, the density p ( s ) for the complete threshold-to-threshold polarizationsimplifies, p ( s ) = 2 ln 1 + √ − s − √ − s . (5.19)21he distribution (5.19) matches exactly the avalanche size distribution of Dhar’sAbelian sandpile model in 1d [40–42]. This is not a coincidence, as we nowexplain.In the sandpile model, the height parameter h i can take only nonnegativeinteger values, and is stable only if h i ∈ { , } for all i . Given a 1d sandpile oflength L with sites labeled 1 to L and some stable initial configuration, a site i is selected at random and a grain of sand is added so that h i → h i + 1. Thetoppling rules of the model are as follows:(i) Find any j such that h j ≥
2, set h j → h j −
2, and h j ± → h j ± + 1.(ii) If h i ≥ i , goto (i).It is useful to add pockets , sites i = 0 and i = L + 1, which we do not consideras part of the sandpile. One of the grains which topples from site 1 or L willfall into a pocket and is lost. We fix h = h L +1 = 0.The set of recurrent states R consists of the L + 1 configurations which have h i = 1 for all but at most one site (where it is 0). It can be shown [30, 40, 41]that:(a) Starting with any configuration in R and adding 1 at any site, the sandpilealgorithm produces a result in R .(b) Under dynamics which consist of adding 1 at a site chosen uniformly atrandom and stabilizing, the uniform distribution on R is invariant.(c) Starting with a recurrent state and adding at site k with h k = 1, the active region where topplings occur is the interval containing k boundedby the closest sites i , possibly pockets, to the left and right of k at which h i = 0. As the boundaries σ L , σ R are excluded from the interval for thetoy model, so are the boundaries where h i = 0 excluded from the activeregion.As we have seen in the previous sections, the evolution under the ZFA is adynamics of moving over- or underlines in the rank diagram. In the negativethreshold configuration of the toy model we encounter three types of sites, thosewith an overbar (corresponding to h = 2 sites), those without a bar ( h = 1sites), and those with an underbar ( h = 0 sites). After aligning the activeregions of both models, which have identical size if we set L = L −
2, we obtaina correspondence between a negative threshold configuration of the toy modeland a recurrent state of the sandpile.We observe also the equivalence of the total threshold-to-threshold evolutionof the toy model and the stabilization of a recurrent sandpile configuration whena single grain is added. The key point is that the totality of the iterated ZFAevolution is Abelian [25]. If we set z +max = max j z + j , the maximum height of thepositive threshold configuration, and then jump all sites which have z j > z +max ,then: • all particles in the active region will be forced to jump at least once,22 the order in which those sites with z j > z +max are jumped is immaterial ifwe are concerned only with the final result, and • the positive threshold configuration is ultimately reached.This is equivalent to running the BTW/Dhar sandpile algorithm on the siteswith overbars, which preserves the correspondence between sandpile and toymodel configurations. However, this map discards the ordering and values ofthe well coordinates z i , which in turn drive the evolution towards threshold inthe ZFA and thereby give rise to a family of distributions (5.12). The toy model ZFA has a very fast numerical implementation which we nowdescribe. For the threshold-to-threshold evolution, the negative threshold con-figurations are generated following (4.11), and the random permutation π from(4.16) is obtained. The evolution proceeds in units of avalanches using Propo-sition 4.3(iii) and the rank representation of configurations, as outlined in thediscussion following the proposition. We therefore only have to keep track of thelocations of the over- and underlines which involves simple integer arithmetic.This implementation is fast, since instead of individual jumps we deal withavalanches and the expected number of avalanches occurring during threshold-to-threshold evolution turns out to scale as ln L , which is what one expects,since a record breaking process underlies the evolution from negative to positivethreshold. An explicit formula for the distribution of the number of steps canbe derived [43].At the end of each avalanche we record various statistics, such as the maxi-mum of z i , the cumulative number of jumps that have occurred at a given site,and the size of the current avalanche. All numerical results presented here wereobtained without parallelization on single processors of an HP Z800 worksta-tion. The longest run of about 262000 realizations of a size L = 131072 systemtook 4 hours.The control parameter for the approach to threshold is the difference betweenthe sample-dependent threshold force F th and the current force F . For the ZFA,which holds the force fixed at 0, the appropriate parameter is X = max i z i − max i z + i = max i z i − ζ π (1) , (6.1)where the last equality follows from (4.18).The values of X are recorded at the end of each avalanche. In the courseof threshold-to-threshold evolution, we obtain a decreasing sequence X τ of X values, where τ indexes the avalanches. Following the definition of the corre-sponding processes, (7.59b), if we want to obtain statistics for a particular value x , the contributing avalanches τ will be those which satisfy X τ ≤ x < X τ − ,since the corresponding configuration driven under an external force could havebeen translated by this amount x without incurring any particle jumps. This is23igure 7: Numerical results for the expected cumulative jump size P ( X ) vs. thereduced and rescaled force u = XL in the evolution to threshold, starting fromflat initial conditions, m = . Colors refer to different system sizes L , as indi-cated in the legend, with accompanying numbers of realizations in parentheses.The dashed line is a power law with exponent − x -dependent avalanche size distributions and their expectation valueshave been obtained in Figures 5 and 6.We have also simulated the evolution from a flat initial configuration, m = ,to positive threshold. The evolution proceeds again by avalanches and Figure 7shows our numerical results. The curves for different system sizes collapse forvalues of u = XL <
10 under the scaling of the axes as indicated in the figure.The scaling of P with L − / at u = 0 is in agreement with the predictionfollowing Corollary 5.4. The scaling of the abscissa as XL / suggests that thecorrelation lengths ξ scales now as ξ ∼ X − . We return to a discussion of thisresult in the conclusion. In this section we provide proofs for the results stated in the preceding text, inorder of appearance.
Proposition 3.2.
That (i) the number of jumps is finite, and in fact bounded by L , is immediate from (ii) m ∗ ≤ m + 1, so we proceed to the latter. We argueinductively: suppose that after some execution of (A4) we have well numbers m ′ and well coordinates ˜ y ′ , and that m ′ ≤ m + 1. If max i ˜ y ′ i ≤ / y ′ k > / k . We claim m ′ k = m k , i.e. site24 has not yet jumped. For the full model, observe that the jump response (3.1)has (cid:20) − η η + 1 − η η η L − η L (cid:21) + L − X i =1 (cid:20) − η η η i + η L − i − η L (cid:21) = 0 . (7.1)It follows that any particle which has jumped has well coordinate at most whatit was after (A3), namely 1 /
2. For the toy model, a site which has jumped oncewith neighbors which have each jumped at most once has no increase beyondits value after (A3). In either case, m ′ k = m k follows, and m ′ + δ k ≤ m + 1.For (iii), if a given site i has not jumped, then ˜ y ∗ i is obtained from ˜ y i > − / F ∗ − F ) /λ and then adding the (positive) effects of jumpsat the other sites. So there is nothing to check unless the site i has jumped. Inthis case, ˜ y i + F ∗ − Fλ + a > + 12 (7.2)with a the (positive) effect of the jumps at other sites which have preceded thejump at i , and ˜ y ∗ i = ˜ y i + F ∗ − Fλ + a − (cid:18) η η − O ( η L ) (cid:19) > − η η = F ∗ − Fλ + 12 − F ∗ − Fλ − η η . So we require F ∗ − Fλ + 2 η η < . (7.3)Since F ≥
0, one can easily verify from (2.8) that the sum of the well coordinates˜ y i is nonnegative regardless of m . Thus values of F ∗ with ( F ∗ − F ) /λ > / F ∗ − F ) /λ ≤ /
2. Thechoice η < / η/ (1 + η ) < /
2. The desired inequality follows.
Lemma 3.4.
Suppose we are applying ZFA to m and that m ′ is either equal to m or an intermediate configuration obtained after some execution of (ZFA3)for which m ≤ m ′ ≤ m . For any j such that m ′ j = m j ,˜ y ′ j = 1 − η η X i ∈ Z η | i − j | ( m ′ i − m ′ j + α i − α j ) ≤ − η η X i ∈ Z η | i − j | ( m i − m j + α i − α j ) = ˜ y j in the case of the full model, and˜ y ′ j = η ( m ′ j − − m ′ j + m ′ j +1 ) ≤ η ( m j − − m j + m j +1 ) = ˜ y j i ˜ y i > max i ˜ y i , then ˜ y ′ j < max i ˜ y i , and site j will not jump. Thusthe next iteration of (ZFA3), if any, will produce m ′′ which still has m ′′ ≤ m .If (ii) max i ˜ y i = max i ˜ y i , then ˜ y ′ j ≤ max ˜ y i and site j will only jump if m ′ = m , i.e. in (ZFA1), and j = arg max i ˜ y i . If m j < m j , this jump does notcause a crossing.Since m ≤ m ∗ and max i ˜ y i ≥ ˜ y ∗ i trivially, and having established (i) and(ii), for (iii) we need only consider the case wheremax i ˜ y i = max i ˜ y ∗ i = max i ˜ y i (7.4)and m j = m j for j = arg max i ˜ y i . As in the proof of (i), we find ˜ y j ≤ ˜ y j , so j = arg max i ˜ y i as well, so that m ∗ j = m j + 1 > m j . Invoking (ii), we aredone. Proposition 3.6. (i) For existence, recall from (2.8) that the well coordinatescan be expressed in terms of the Laplacians ∆ m and ∆ α . We know that P L − i =0 ∆ m i = 0, so large negative values of ∆ m i will require also large positivevalues elsewhere. The equation (2.3) (at F = 0) can be rewritten as λ ˜ y i = ∆ m i + ∆ α i + ∆˜ y i . (7.5)Noting that | ∆ α i | and | ∆˜ y i | are bounded by 2, we see that large positive valuesof ∆ m i will cause large positive values of ˜ y i . We may therefore optimize over∆ m uniformly bounded above by 8 (since anything above this is guaranteed tobe worse than taking m = 0) and thus below by 8 L , and there are only finitelymany possibilities. Existence is immediate.Uniqueness requires separate arguments for the full model and the toy model.For the full model, suppose we have m and m threshold configurationswhich do not differ by simple translation. Since overall translation does notaffect Laplacians, it doesn’t affect well coordinates, so we may as well assumemin i m i − m i = 0 and m = m .We first argue that it suffices to consider m ≤ m + . Apply the ZFA to m , producing m ∗ , also a threshold configuration, and m ∗ ≤ m + . Write j = arg max i ˜ y i . If m j < m j , we have m ∗ ≤ m by Lemma 3.4. We rule out m j = m j because it forces ˜ y j > ˜ y j = max i ˜ y i , (7.6)in which case m is not a threshold configuration. Thus m ∗ is a thresholdconfiguration which has m ≤ m ∗ ≤ m + and m j < m ∗ j . We may as wellassume that m has these properties.Let j = arg max i ˜ y i . Since m j < m j and m ≤ m + , we have ˜ y j > ˜ y j ,so j = j . Now consider the underlying randomness α : for ˜ y j = ˜ y j we musthave, using (2.8), X i ∈ Z ( η | i − j | − η | i − j | )∆ α i = X i ∈ Z η | i − j | ∆ m i − η | i − j | ∆ m i . (7.7)26ecalling that we’re dealing with a periodic system, so that the above may bereplaced with a finite sum, and that for threshold configurations the number ofpossible values for ∆ m is finite, we see that ˜ y j = ˜ y j requires that a nonde-generate linear functional of α takes one of finitely many values, which happenswith probability zero.We turn to uniqueness for the toy model. Again take threshold configurations m and m with m = m and min i m i − m i = 0. As we did for the full model,we begin by reducing the class of m we must consider. Write m ∗ for the resultof the ZFA applied to m . If m j < m j , then m ∗ ≤ m , as desired. On theother hand, m j = m j leads to a contradiction: let ℓ and r be the first indicesto the left and right, respectively, of j for which m ℓ > m ℓ and m r > m r . Usingthe formula z i = ∆ m i + ∆ α i , we see that z ℓ +1 ≥ z ℓ +1 + 1 and z r − ≥ z r − + 1 , (7.8)and it follows that z ℓ +1 + 1 and z r − + 1 are both less than max i z i . For reasonsas in the uniqueness argument for the full model, this inequality is almost surelystrict. Define m ′ by m ′ i = m i + ( i − ℓ − + − ( i − j ) + − ( i − j + r − ℓ ) + + ( i − r − + . (7.9)Then z ′ differs from z in only four locations, ℓ + 1, j , j + r − ℓ , and r −
1, with z ′ i − z i having values +1, − −
1, and +1, respectively. Then max i z ′ i < max i z i follows from (7.8), which is a contradiction.Thus it suffices to take m = m ∗ , and show that the assumption min i m ∗ i − m i = 0 leads to a contradiction. When we apply the ZFA to m , the site j =arg max i m i will jump. If min i m ∗ i − m i = 0, not all sites jump. Letting ℓ and r be as above, we can again construct m ′ with max i z ′ i < max i z i , contradictingoptimality and finishing the proof of uniqueness.For (ii), take a threshold configuration m + and translate it so that min i m + i =0. Starting with m = 0, repeatedly apply ZFA. By Lemma 3.4, the sequence of m produced cannot cross m + unless we obtain m so that max i ˜ y i = max i ˜ y + i ,that is, another threshold configuration. On the other hand, we must jump atleast once with each ZFA application, so crossing m + after finitely many stepsis unavoidable.Part (iii) is immediate from (i) and Proposition 3.2.To verify that the description of the threshold configuration given by (4.3)in Theorem 4.1 gives a legitimate vector m + of well numbers, we require thefollowing elementary lemma. Lemma 7.1.
A vector ℓ ∈ Z L is equal to ∆ m for some m ∈ Z L if and only ifboth of the following hold:(i) P L − i =0 ℓ i = 0 (ii) P L − i =0 iℓ i ≡ L ) 27 roof. That ∆ on Q L with periodic boundary is self-adjoint, together withstandard linear algebra (namely the identification of the cokernel with the or-thogonal complement of the range) shows that condition (i) is necessary andsufficient for ∆ m = ℓ to have a solution m ∈ Q L . The only question is whetherthere is a solution with integer entries. For this it is necessary and sufficientthat a solution m ∈ Q L have m − m ∈ Z . Necessity is obvious and sufficiencyfollows if we set m = 0, m according to the known difference m − m , andrepeatedly use m i +1 = − m i − + 2 m i + ℓ i to obtain the other entries, which willbe integers.An easy induction shows that for k ≥ m k = − ( k − m + km + k X i =1 ( k − i ) ℓ i . (7.10)Setting k = L in the above, recalling m = m L , and rearranging we find L ( m − m ) = L X i =1 ( L − i ) ℓ i . (7.11)From this, we see m − m ∈ Z if and only if P Li =1 ( L − i ) ℓ i is a multiple of L ,which is easily shown to be equivalent to (ii). Theorem 4.1.
Lemma 7.1 guarantees that the specification given for ∆ m + isadmissible, i.e. can be inverted to obtain m + ∈ Z L . To verify the optimality of m + , we invoke Proposition 3.6, claiming that the ZFA applied to m + produces m + + .We claim that z + i + 1 > z +max for all i = k + and z + k + + 2 > z +max . Since ajump at site i increases z i ± by 1, each jump that occurs, starting at arg max i z + i ,causes both its neighbors to jump except possibly if one of those neighbors issite k + . Due to periodicity, both k + ± z + k + by 2, and itmust jump as well. Verifying the claim will prove the theorem.Using the notation of (4.6), z + i = ω i + P Sj =0 δ iσ ( j ) − δ ik + if S ≥ − P | S |− j =1 δ iσ ( L − j ) − δ ik + if S < ω i ∈ ( − , + ). Suppose S >
0. If i ∈ { σ (0) , . . . , σ ( S ) } \ { k + } , then z + i + 1 = ( ω i + 1) + 1 > z +max , (7.13)and if i ∈ { σ ( S + 1) , . . . , σ ( L − } \ { k + } , then z + i + 1 = ω i + 1 > z + σ ( S − ∨ z + σ ( S ) = z +max . (7.14)If k + ∈ { σ (0) , . . . , σ ( S ) } then z + k + + 2 = ω k + + 2 > z + σ ( S − ∨ z + σ ( S ) = z +max , (7.15)28ut if k + ∈ { σ ( S + 1) , . . . , σ ( L − } then z + k + + 2 = ω k + + 1 > z + σ ( S − ∨ z + σ ( S ) = z +max . (7.16)We omit the verification in the cases S = 0 and S <
0, these being similarexercises in checking cases.
Proposition 4.3.
Note first that (4.14e) follows once we’ve established the cor-responding changes to z , as the change in m has the correct Laplacian, andminimum 0.We argue by induction on ℓ . When ℓ = 1, we are only characterizing theresult of a single iteration of the ZFA. It is straightforward to verify that the setof sites that jumps is W : the site i itself makes the first jump, increasing theheight of each neighbor i ± z i ± + 1 > z i .This outward moving wave terminates when the sites i L + 1 and i R − i R − − ( i L + 1) + 1 = i R − i L − . Since ℓ = 1, we know that either i − i L = 1 or i R − i = 1; by symmetry, we mayas well assume the former. Then( i − i L )( i R − i ) = i R − i = i R − ( i L + 1) , and (i) is satisfied. For (ii), we observe that the result of an interval of sitesjumping is as follows: i L i R jumps : · · · · · · · · · change in z : · · · − · · · − · · · We see that z i L → z i L + 1 and z i R → z i R + 1, as needed. Again assuming that i − i L = 1, we find z i → z i −
1. Since we have i L + i R − i = i R − , so z i L + i R − i → z i L + i R − i − ℓ >
1, assuming the result holds for smaller values.Following the same reasoning as above, the sites W jump in the first applicationof the ZFA, but now i − i L > i R − i >
1, so both i ± z i is unchanged, hence still the maximum. Regarding the configurationafter jumping sites W as the new starting point, we have the same i and z i ,and i ′ L = i L + 1 and i ′ R = i R −
1. Applying the inductive hypothesis, we knowthe effect of iterations 2 , . . . , ℓ . The number of jumps is therefore[( i R − − ( i L + 1) + 1] + ( i − i ′ L )( i ′ R − i ) = ( i − i L )( i R − i ) . z for iteration 1 consist of z i L → z i L + 1 z i R → z i R + 1 (7.17) z i L +1 → z i L +1 − z i R − → z i R − − . (7.18)The changes due to iterations 2 , . . . , ℓ are z i L +1 → z i L +1 + 1 z i R − → z i R − + 1 (7.19) z i → z i − z i ′ L + i ′ R − i → z i ′ L + i ′ R − i − . (7.20)Noting i ′ L + i ′ R − i = i L + i R − i , we see that (7.17) and (7.20) are the desiredchanges and that (7.18) and (7.19) cancel. Proposition 5.1.
We begin by describing the distribution claimed for ω in greaterdetail. For the uniform surface measure on the intersection of the cube ( − , + ) L with the planes P L − n =0 b n ∈ Z , a consequence of | b n | < is that this surface canbe recognized as the graph of a function: b L − = g ( b , . . . , b L − ) ≡ r P L − n =0 b n z − P L − n =0 b n (7.21)is immediate from b L − + P L − n =0 b n = r P L − n =0 b n z , which is forced since theleft-hand side is exactly an integer, and since | b L − | < , it must be the integernearest P L − n =0 b n . The function g has constant gradient ( − , . . . , −
1) where thegradient exists, and it fails to exist only on the ( L − ( ( b , . . . , b L − ) : L − X n =0 b n ∈
12 + Z ) . We therefore recognize the law of { β n } L − n =0 as the result of taking { β n } L − n =0 to be i.i.d. uniform and pushing this measure forward onto the graph of g .This facilitates the following calculation, for trigonometric polynomials f n ( t ) = P | k |≤ K ˆ f n ( k ) exp(2 πikt ), K an arbitrary positive integer: E L − Y n =0 f n ( β n ) = X | k n |≤ K L − Y n =0 ˆ f n ( k n ) ! E exp[2 πik · β ] , (7.22)and E exp[2 πik · β ]= Z ( − , + ) L − exp ( πi " L − X n =0 k n b n + k L − J L − X n =0 b n K − L − X n =0 b n ! db · · · db L − = Z ( − , + ) L − exp ( πi " L − X n =0 ( k n − k L − ) b n + k L − J L − X n =0 b n K db · · · db L − = Z ( − , + ) L − exp ( πi " L − X n =0 ( k n − k L − ) b n db · · · db L − = ( k = · · · = k L − ) . ω i is the representative in (cid:0) − , + (cid:1) of the equivalence class of∆ α i (mod 1), so it will suffice to understand the law of 1-periodic functions of { ∆ α i } . With trigonometric polynomials f n as before, we compute E L − Y n =0 f n (∆ α n ) = X | k n |≤ K L − Y n =0 ˆ f n ( k n ) ! E exp[2 πik · ∆ α ] , (7.23)the summation over integer vectors k with all components bounded by K , and E exp[2 πik · ∆ α ] = E exp[2 πi ∆ k · α ] = L − Y n =0 E exp[2 πi ∆ k n α n ]= (∆ k = 0) = ( k = · · · = k L − ) , since the kernel of the periodic Laplacian consists of constant vectors. Then(7.23) simplifies as E L − Y n =0 f n (∆ α n ) = X | k |≤ K L − Y n =0 ˆ f n ( k ) (7.24)where k is now a single integer (corresponding to a vector with components k n which are identical).Thus E L − Y n =0 f n ( β n ) = X | k |≤ K L − Y n =0 ˆ f n ( k ) = E L − Y n =0 f n (∆ α n ) , (7.25)and by Stone-Weierstrass we extend to general 1-periodic functions f n as neededto verify the proposition. Corollary 5.2.
Using Proposition 5.1, we have S = L − X i =0 J ∆ α i K = L − X i =0 ∆ α i − ω i = − L − X i =0 ω i (7.26)for ω , . . . , ω L − i.i.d. with mean 0 and variance and | ω L − | < . Thestandard central limit theorem then gives (i). The number of topological defectsis one of | S | or | S | + 2, and (ii) is immediate.The exchangeability claimed in Theorem 5.3 requires a more detailed exam-ination of the threshold configuration. We begin by noting the formula for ∆ m at ( ± )-threshold (4.3) can be viewed as a result of applying two corrections tothe − J ∆ α K sequence:∆ m i = − J ∆ α i K + J ′ i + ( δ iℓ + − δ iℓ − ) , (7.27)31here J ′ i = − ( i ∈ σ { L − | S | , . . . , L − } ) if S <
00 if S = 0 ( i ∈ σ { , . . . , S − } ) if S > ℓ ± are selected as follows: for the (+)-threshold configuration, we set ℓ + = σ ( L − | S | ) if S < σ (0) if S = 0 σ ( S ) if S > − )-threshold configuration, we set ℓ − = σ ( L − | S | −
1) if
S < σ ( L −
1) if S = 0 σ ( S −
1) if
S > . (7.30)In both cases, the choice of ℓ ± dictates a corresponding ℓ ∓ via the L -divisibilitycondition of Lemma 7.1. We thus view the ( ± )-threshold configurations as “oneup, one down” perturbations of − J ∆ α K + J ′ , with the same spacing d ≡ ℓ + − ℓ − (mod L ) = L − X i =0 i ( J ∆ α i K − J ′ i ) (mod L ) (7.31)between the ±
1, and we insist on choosing ℓ ± for the ( ± )-threshold, respectively.The location of the negative defect in the ( − )-threshold is important forthe threshold-to-threshold evolution, and, in light of the above, this amounts tounderstanding d and σ . For this, and the exchangeability result Theorem 5.3,we need to understand the relationship between d and ω . Fortunately theseinteract as nicely as one could hope. Lemma 7.2.
The difference d between ℓ ± defined by (7.31) is uniform on { , . . . , L − } and independent of ω .Proof. We begin with the part of d which depends on J ∆ α K , claiming that L − X i =0 i J ∆ α i K (mod L ) (7.32)is uniform on { , . . . , L − } and independent of ω .For independence from ω , it is sufficient to consider { ω i } L − i =1 , since ω is afunction of these. We have L − X i =0 i J ∆ α i K = L − X i =0 i (∆ α i − ω i ) = L ( α − α L − ) − L − X i =0 iω i , (7.33)32nd claim that { α − α L − mod 1 , ω , . . . , ω L − } are distributed as i.i.d. uniform(mod 1) variates conditioned to have L ( α − α L − ) − L − X i =1 iω i ∈ Z . (7.34)We calculate in the manner of Proposition 5.1. For f n ( t ) = P | k |≤ K ˆ f n ( k ) exp(2 πikt ),consider E f ( α − α L − ) Q L − n =1 f n (∆ α n ): X | k n |≤ K L − Y n =0 ˆ f n ( k n ) E exp[2 πik · ( α − α L − , ∆ α , . . . , ∆ α L − )] . (7.35)Write A for the matrix mapping ( α , . . . , α L − ) ( α − α L − , ∆ α , . . . , ∆ α L − ).We need to evaluate E exp[2 πi k · A α ] = E exp[2 πiA T k · α ] = ( A T k = 0) , (7.36)and therefore require a description of ker A T . We have A = · · · − − · · · − · · · · · · − · · · − , (7.37)and see that A T has rows 2 through L − L −
1) in commonwith the Laplacian; that (∆ k , . . . , ∆ k L − ) = means ( k , . . . , k L − ) is flat, sothat ( k , . . . , k L − ) = ( an + b ) L − n =1 (7.38)for some constants a and b . The second row then gives0 = − a + b ) + 1(2 a + b ) = − b. (7.39)The first row gives 0 = k + 1 a + ( L − a = k + La, (7.40)and the last 0 = − ( − La ) + ( L − a − L − a = 0 (7.41)imposes no additional constraint. Thus A T k = if and only if k = ( k , . . . , k L − ) = ( − La, a, a, . . . , ( L − a ) (7.42)for some constant a . 33ompare this with the following: let β , . . . , β L − be i.i.d. uniform ( − , + ), θ ∈ { , . . . , L − } uniform and independent of the β i , and γ = 1 L θ + L − X n =1 nβ n ! (mod 1) . (7.43)For f , . . . , f L − as before, we compute E f ( γ ) L − Y n =1 f n ( β n ) = X | k n |≤ K L − Y n =0 ˆ f n ( k n ) E exp[2 πik · ( γ, β , . . . , β L − )] . (7.44)Here E exp[2 πik · ( γ, β , . . . , β L − )]= E exp ( πi " k L θ + L − X n =1 nβ n ! + L − X n =1 k n β n = E exp ( πi " k θL + L − X n =1 (cid:18) nk L + k n (cid:19) β n = L L − X t =0 e πik t/L ! E exp ( πi L − X n =1 (cid:18) nk L + k n (cid:19) β n ) . Note that e πik /L is an L th root of unity, so the left sum above is zero unless L divides k , in which case the sum is L . But if L divides k , say k = − La , then E exp ( πi L − X n =1 (cid:18) nk L + k n (cid:19) β n ) = (cid:18) k n = − nk L for n = 1 , . . . , L − (cid:19) , (7.45)which can be nonzero only if k n = − n ( − La ) /L = na for n = 1 , . . . , L −
1. Thus { γ, β , . . . , β L − } d = { α − α L − (mod 1) , ω , . . . , ω L − } . (7.46)Now that we know P L − i =0 i J ∆ α i K is independent of ω , and that J ′ is a func-tion of ω , we use the following elementary fact: if X and Y are independentrandom variables in Z /L Z and Y is uniform, then X + Y is uniform and inde-pendent of X . Independence of d and ω is immediate. Theorem 5.3.
Exchangeability of the components ω i is immediate from Propo-sition 5.1. We have z + i = ∆ m i + ∆ α i = ω i + J ′ i + δ iℓ + − δ iℓ − . (7.47)By construction (7.28) and (7.29), J ′ i and δ iℓ + are functions of the value ω i andthe unordered set of values { ω , . . . , ω L − } . Using the preceding Lemma 7.2, wefind ℓ − = ℓ + − d is uniform on { , . . . , L − } and independent of ω .34e then recognize z + i given by (7.47) as a function of ω i , the set of values { ω , . . . , ω L − } , and ℓ − , the last of which is independent of ω . Exchangeabilityof the components of z + follows. Corollary 5.4.
We first use Theorem 5.3 and a standard result (see for example[44, Thm. 24.2] or [37, Thm. 16.23]) to show that the processesˆ s ( L ) ( t ) ≡ L − / ⌊ Lt ⌋ X i =0 z + i (0 ≤ t ≤
1) (7.48)converge in distribution in the Skorokhod space D ([0 , − / B ( t ) where B ( t ) is standard Brownian bridge. We claim that we have distributional con-vergence, L − / L − X i =0 z + i , L − L − X i =0 ( z + i ) δ L − / z + i ! d → (0 , (12 L ) − δ ) ∈ R ×M ( R ) , (7.49)where M ( R ) is the space of locally finite measures on R equipped with the vaguetopology. In fact, the first component is exactly equal to 0, so we focus on thesecond component, which we write as L − L − X i =0 ( z + i ) δ + L − L − X i =0 ( z + i ) ( δ L − / z + i − δ ) . (7.50)We claim the second sum above can be ignored as L → ∞ . Fix a continuous,compactly supported function f on R , and any ǫ >
0. Choose L sufficientlylarge that | x | < L − / implies | f ( x ) − f (0) | < ǫ , and observe that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z f ( x ) L − L − X i =0 ( z + i ) ( δ L − / z + i − δ )( dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ | z + i | ≤ . Distributional convergence of the first sum ofmeasures in (7.50) amounts to distributional convergence of the coefficient L − L − X i =0 ( z + i ) = L − L − X i =0 ( ω i + z + i − ω i ) = L − L − X i =0 ω i + L − L − X i =0 ( z + − ω i )( z + i + ω i ) d → . Here we have used the (weak) law of large numbers on P L − i =0 ω i , since removingone term restores independence, and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L − L − X i =0 ( z + − ω i )( z + + ω i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ L − L − X i =0 | J ′ i + δ iℓ + − δ iℓ − | (2)= 2 L − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L − X i =0 ω i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d → s ( L ) ( t ) follows.We now return to s ( L ) ( t ). Writing ˆ s i ≡ P ij =0 z + i , a routine calculation gives s i − ˆ s i − L L − X j =0 ˆ s j = α i − α i +1 . (7.52)In particular, the difference on the left-hand side is bounded by a constant, andthus disappears in the central limit scaling. Note also that1 L L − X j =0 ˆ s j = Z ˆ s ⌊ t/L ⌋ dt, (7.53)and that integration R · dt is a continuous functional on the Skorokhod space D ([0 , B ( t ) has mean zero is immediate, and that it is Gaussian follows fromeasy arguments. The discrete analogue, a Gaussian vector with its sum sub-tracted from each component, is of course standard, since (possibly degenerate)Gaussian distributions are preserved under affine maps. Working on the levelof continuous processes, we can fix some 0 = t < t < · · · < t n − < t n = 1 andobserve using standard properties of Brownian bridge that Z B ( r ) dr − n X i =1
12 [ B ( t i − ) + B ( t i )]( t i − t i − ) (7.54)is Gaussian and independent of ( B ( t ) , . . . , B ( t n )).Stationarity can be deduced from that of the sequence of strains s i , or fromcomputing the covariance E B ( t ) B ( t ′ ) for some t, t ′ ∈ [0 ,
1] and recognizingthis as a function of the difference t ′ − t ; recall that wide-sense stationarity andstationarity are equivalent for Gaussian processes. The formula (5.4) is obtainedusing Fubini’s theorem and calculus. Proposition 5.5.
By the apparent exchangeability of the components of ζ , theunordered pair { π L , π R } is uniformly distributed over all pairs of (mod L equiv-alence classes) of indices. Using Lemma 7.2 as in the proof of Theorem 5.3,we find that k − ranges over all indices and independent of ζ , and thus alsoindependent of { π L , π R } .As in the discussion surrounding (4.23), we may by translation assume that k − = 0, and select the representatives of π L and π R which satisfy π R − L < π L ≤ ≤ π R < π L + L. (7.55)By the discussion above, we find ( π L , π R ) is uniformly distributed over the set { ( i, j ) ∈ Z : i ≤ , j ≥ , i = j, | i | + j < L } , (7.56)36nd independent of the unordered set of values { ω , . . . , ω L − } or { ζ , . . . , ζ L − } .The set in (7.56) has cardinality ( L + 2)( L − / either π L = 0 or π R = 0, the threshold-to-thresholdpolarization is zero because the ( ± )-threshold configurations are the same. Thisfollows using (4.19), which says that if k − is one of π L or π R , then k + is the other,and the formulas (4.17) and (4.18) match. Note that these cases contribute abounded (in fact, zero) quantity to the polarization, and occur with probabilityonly of order O ( L − ). In the limit as L → ∞ , this can be ignored. We replace(7.56) with { ( i, j ) ∈ Z : i ≤ − , j ≥ , | i | + j < L } , (7.57)which has cardinality (cid:0) L (cid:1) .As stated, we will now make the assumption that the genuine situation canbe approximated by ζ π (0) = ζ π (1) = − / J L and J R of(4.21) and (4.22) can be approximated by i.i.d. uniform ( − , + ) variables in-dependent of π (0) and π ( L ). Following this assumption, the calculation is exact.Subtracting − / ,
1) variables X π L +1 , . . . , X , . . . , X π R − . (7.58)We might extend this to a bi-infinite sequence of i.i.d. uniform (0 ,
1) variates,and then define for 0 ≤ x ≤ j ′ L ( x ) = max { i ≤ X i ≤ x } j ′ R ( x ) = min { j ≥ X j ≤ x } (7.59a) j L ( x ) = j ′ L ( x ) ∨ π L j R ( x ) = j ′ R ( x ) ∧ π R . (7.59b)Recalling the cumulative avalanche size Σ( x ) and polarization P ( x ) are givenby LP ( x ) = Σ( x ) = − j L ( x ) j R ( x ), we wish to characterize the distribution ofthe pair ( j L ( x ) , j R ( x )).The distribution of ( j L ( x ) , j R ( x )) can be computed precisely on the discretelevel, but since we are interested in the behavior as L → ∞ , we may as wellrescale and pass to continuous variates. We claim that as L → ∞ , for fixed u ≥ (cid:18) − j ′ L ( u/L ) L , j ′ R ( u/L ) L , − π L L , π R L (cid:19) d → ( γ ′ L ( u ) , γ ′ R ( u ) , ρ L , ρ R ) , (7.60)where γ ′ L ( u ) and γ ′ R ( u ) are independent exponential random variables with rate u , and are independent from the pair ( ρ L , ρ R ), which is uniformly distributedon the triangular region with vertices (0 , , , j ′ L ( x ) and j ′ R ( x ) are conditionally independent onthe event j ′ R ( x ) >
0. For fixed u , as L → ∞ , the probability that j ′ R ( u/L ) = 0tends to zero. Observe also that for fixed u , P (cid:18) j ′ R ( u/L ) L ≤ t (cid:19) = ⌊ Lt ⌋ X n =0 (cid:16) − uL (cid:17) n uL = 1 − (cid:16) − uL (cid:17) ⌊ Lt ⌋ +1 → − e − ut , (7.61)37ointwise for all t . Finally, that ( − π L /L, π R /L ) converges distributionally to( ρ L , ρ R ) is immediate, since computing an expectation of some function withrespect to the law of ( − π L /L, π R /L ) is more or less a Riemann sum for theintegral over the triangle.Since ( a, b, c, d ) ( a ∧ c, b ∧ d ) is continuous, and distributional convergenceis preserved under continuous maps, it follows that (cid:18) − j L ( u/L ) L , j R ( u/L ) L (cid:19) = (cid:18) − j ′ L ( u/L ) L ∧ − π L L , j ′ R ( u/L ) L ∧ π R L (cid:19) d → ( γ ′ L ( u ) ∧ ρ L , γ ′ R ( u ) ∧ ρ R ) ≡ ( γ L ( u ) , γ R ( u )) . We address the limiting statistics of the former using a calculation with thelatter, continuous variates.Rescaling Σ( x ) as ς ( u ) ≡ lim L →∞ Σ( u/L ) /L , (7.62)we obtain the density of ς ( u ), which we denote p u ( s ) = P ( ς ( u ) ∈ ds ) /ds , p u ( s ) = Z d x Z − x d y δ ( s − xy ) e − u ( x + y ) (cid:2) u (1 − x − y ) + u (1 − x − y ) (cid:3) , (7.63)where δ ( x ) is the Dirac delta. Carrying out next the integration over y , makingthen a change of variable z = x + s/x in the remaining integral, and taking careof the integration boundaries we obtain p u ( s ) = Z √ s d z e − zu u (1 − z ) + 2 u (1 − z ) ( z − s ) / , (7.64)which is supported on [0 , ]. This is the formula claimed in (5.12). The ex-pectation value of ς ( u ), (5.11), then follows from the distribution p u ( s ) by anexchange of the order of integration and some repeated integration by parts. The CDW toy model introduced in [25] and studied in this article exhibits acritical depinning transition. It retains similarities with the untruncated CDWmodel, while admitting some explicit formulas which make rigorous analysispossible. However, it does not appear to be completely trivial. Our understand-ing of the threshold-to-threshold evolution is rather complete, as the changesare confined to a single active region growing in a simple way, but the flat-to-threshold evolution has so far resisted nice analytical characterizations. Insimulations we see multiple regions of activity, which grow and merge. This canbe understood by noting that the initial well-coordinates are distributed withinan interval of width larger than 1. The evolution towards positive thresholdvia the ZFA, while conserving the fractional part of the well-coordinates, grad-ually reduces this width by successively pruning the integer parts of the well38oordinates. This means that while avalanches terminate at sites with low well-coordinates, these values are often so low, that their increments by +1 at theavalanche termination, as prescribed by Proposition 4.3 (ii) , will not make themavalanche initiation sites for the next avalanche. Rather, there will be other siteswith higher z -values that serve as avalanche initiators. This situation will con-tinue until such sites have been depleted sufficiently that the termination sitesof the previous avalanches do initiate the next avalanche. This is the majordifference from the threshold-to-threshold evolution where—due to the natureof the initial configuration—this termination/initiation pattern is observed im-mediately from the start. It was this observation that allowed for a descriptionin terms of a record-breaking process. The behavior of the evolution startingform a flat initial configuration is more interesting, but also more difficult todescribe precisely.Another set of interesting questions relate to hysteretic behavior as the forceis raised and lowered, a feature previously observed in CDW simulations [23].For his recent master’s thesis, Terzi [43] studied numerically hysteresis in thetoy model. In the toy model this occurs when the external force goes through asequence of force increments and decrements after which it returns to its initialvalue. In terms of the ZFA evolution this amounts to running this algorithm inthe backward direction: Algorithm 3.3 with obvious modifications correspondingto force decrements. Starting from a ( ± )-threshold configuration and applyingsequences of forward and backwards steps of the ZFA, Terzi finds that the totalnumber of reachable configurations scales like L / . One might hope that forthe toy model, this can be shown analytically, but this is not yet done. Terzihas additionally shown that the hysteretic behavior of the toy model exhibitsthe return point memory effect. This is a direct consequence of the no crossingproperty of the evolution [45], which for our model is guaranteed by Lemma 3.4.The approach to the depinning transition using renormalization group ideas[15–17] suggests universality of the behavior, near the transition. In particularit is believed that the values of the scaling exponents should depend little on themicroscopic details of the underlying model. The toy model serves as a goodexample to test these assumptions. Here we find that depending on the initialconfiguration chosen, the evolution to threshold and the corresponding scalingbehavior is markedly different. While in the threshold-to-threshold evolutionthe correlation length near threshold diverges as ξ ∼ X − and the quantity thatexhibits scaling is the cumulative avalanche size Σ which scales as Σ ∼ X − ,in the evolution from a flat initial condition to threshold we find numericalresults consistent with ξ ∼ X − and P ∼ X − , which moreover agrees withthe renormalization group based prediction of Narayan et al. , [15, 24]. The toymodel illustrates that the choice of initial condition can result in dramaticallydifferent dynamics, leading to these disparate exponents.Another type of universality is the robustness of our results when we changethe law of the underlying disorder. Theorem 4.1 can address immediately anyrandomness which is mutually absolutely continuous with the α considered hereand thus one can ask whether scaling still holds, and if so, how the scalingexponents characterizing the correlation length and the cumulative avalanche39izes, depend on the probability laws for the underlying disorder.Generalization to higher dimensions is a more serious undertaking, as thetraditional two-dimensional sandpile is already much more intricate than its one-dimensional relative [30]. On the other hand, the randomness could conceivablybe helpful: the size of the set of recurrent sandpile configurations should besmaller if the sites on the lattice are no longer identical. The authors hope toconsider this matter, and others mentioned above, in future work. Acknowledgements
DK and MM thank F. Rezakhanlou for stimulating discussions. DK acknowl-edges the hospitality of the Istanbul Center for Mathematical Sciences (IMBM)and the Mathematics and Physics Departments of Bo˘gazi¸ci University. MMacknowledges discussions with H.J. Jensen, M.M. Terzi, P.B. Littlewood, S.N.Coppersmith and A. Bovier. He thanks the Berkeley Math department for theirkind hospitality during his sabbatical stay. This work was supported in part byby NSF grant DMS-1106526 and Bo˘gazi¸ci University grant 12B03P4.
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