Interface depinning versus absorbing-state phase transitions

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Interface depinning versus absorbing-state phase transitions

Mikko Alava 1 and Miguel A. Munoz2

1 Helsinki University of Technology, Lab. of Physics, HUT-02105 Finland2 Instituto de Fısica Teorica y Computacional, Carlos I, Universidad de Granada, Facultad de Ciencias, 18071-Granada,

Spain.

(June 17, 2011)

According to recent numerical results from lattice models, the critical exponents of systemswith many absorbing states and an order parameter coupled to a non-diffusive conserved fieldcoincide with those of the linear interface depinning model within computational accuracy. In thispaper the connection between absorbing state phase transitions and interface pinning in quencheddisordered media is investigated. For that, we present a mapping of the interface dynamics in adisordered medium into a Langevin equation for the active-site density and show that a Reggeon-field-theory like description, coupled to an additional non-diffusive conserved field, appears rathernaturally. Reciprocally, we construct a mapping from a discrete model belonging in the absorbingstate with-a-conserved-field class to a discrete interface equation, and show how a quenched disorderis originated. We discuss the character of the possible noise terms in both representations, andoverview the critical exponent relations. Evidence is provided that, at least for dimensions largerthat one, both universality classes are just two different representations of the same underlyingphysics.

I. INTRODUCTION

Phase transitions separating a non-trivial from a frozenphase, in which the dynamics is completely arrested, ap-pear in a large variety of situations in physics, as well asin many other disciplines [1–3]. A central problem froma theoretical viewpoint is to understand how the symme-tries and conservation laws of the dynamics are reflectedin the categorization of models into universality classes.There are two main general contexts in which this typeof frozen states appear:

(i) Lattice models with discrete particles; typically par-ticles originate “activity” and the frozen state, withoutactivity is referred to as “absorbing state” [1–3]. Thisgroup appears in various disguises as cellular automata[4], reaction-diffusion systems [1,3], directed-percolation-type models [3], or the fixed energy ensemble of sandpilecellular automata [5], among many other examples.

(ii) Elastic interfaces in random environments. In thissecond group, the dynamics is frozen whenever the inter-face is pinned by the disorder, while the non-trivial phaseis the moving or depinned one [6,7].

The number of physical realizations of both of thesetwo generic families of phase transitions is huge [1–3,6,7].

The most prototypical universality class in the firstgroup is that embracing, among many other models andsystems, directed percolation (DP) [1–4]. At a contin-uous level the DP class is represented by the ReggeonField Theory (RFT) [8], which can be written in termsof the following Langevin equation:

∂tρ(x, t) = aρ− bρ2 + ∇2ρ+ σ√ρ η(x, t) (1)

where ρ is an activity field, a, b, and σ are constants andη is a delta-correlated Gaussian white noise. The RFT is

the minimal field theory capturing the relevant ingredi-ents of the DP universality class. It can be renormalizedusing standard field theoretical methods and the associ-ated critical exponents can be computed in ǫ-expansion[8]. Other universality classes of absorbing-state phasetransitions have been identified; all of them owe their ex-istence to the presence of some additional symmetry orconservation law. Among them some example are: theconserved parity (CP) class, in which there are two Z2-symmetric equivalent absorbing states [9,3], dynamicalpercolation [10], and the different classes of transitionswith extra conservation laws [11–13].

In the group of pinned interfaces, the simplest con-tinuous model for depinning is the quenched Edwards-Wilkinson (QEW) equation, also called, “Linear Inter-face Model” (LIM) [6,7]

∂th(x, t) = ν∇2h(x, t) + F + η(x, h) , (2)

that describes an elastic interface (the Laplacian) at thereference height h(x, t), with surface tension ν, underthe influence of a constant external driving term F , anda quenched noise η. Equation (2) exhibits a depinningtransition at a critical force Fc; the interface configu-ration and dynamics develop critical correlations in thevicinity of the critical point. The standard approach for atheoretical analysis of the LIM is the functional renormal-ization group method. One-loop expressions for the mini-mal set of exponents have been computed by Nattermannet al. [14] on one hand, and by Narayan and Fisher [15](see also the more recent work by Le Doussal and collabo-rators [16]). Here one enters technically and conceptuallydifficult terrain due to the renormalization of the wholedisorder correlator. The outcome is that for noise fields η

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which do not exhibit extra translational symmetries, theexpected depinning behavior follows, very generally, thatresulting from a random-field uncorrelated noise term:the LIM universality class [14,15,17,18]. Other univer-sality classes in the interfaces-in-random-media realm arethe quenched Kardar-Parisi-Zhang equation [19,6,7] andthe Edwards-Wilkinson equation with columnar noise[20,7].

Recent investigations (motivated by the analysis ofsandpile models [21,22], the archetype of systems exhibit-ing self-organized criticality (SOC) [23]) have demon-strated that different models showing a continuous transi-tion into an absorbing phase and with an order parametercoupled linearly to an extra, non-diffusive conserved field(NDCF) belong to a unique universality class [13,24,25],that we will refer to as NDCF class. This class differsfrom the extremely robust DP class owing to the pres-ence of an additional conservation law [12]. Moreover,the critical exponents of this class seem, within numeri-cal accuracy, equal to those of the LIM class [5,13,24,25].This might be surprising at first sight, as in these modelsthere is no quenched disorder, as there is in LIM, anddisorder is usually a relevant perturbation when it comesto universality issues.

From a different perspective this observation is not sosurprising, as different often tentative results have beenreported in the literature in order to relate the dynamicsof sandpiles to that of elastic manifolds in random media[26,27]. Furthermore, there is one more viewpoint fromwhich the coincidence between both types of models isnot so striking, namely, that provided by the “Run-TimeStatistics” technique [28]. This technique or theory es-tablishes that quenched disorder can be mapped rathergenerically into long-range temporal correlations (i.e. along-term memory) in the activity field, (note the ideaworks also the other way around) [30], and has been re-cently applied with success to the Bak-Sneppen modelamong others [29]. In the NDCF class the presence ofa conserved field plays the role of a long-memory term,and therefore it comes not as a big surprise that it isequivalent to a class with quenched disorder.

In this article we discuss in detail the relation betweenthe two presented groups of transitions, i.e. absorb-ing states with a conserved field and pinned interfacesin random media, including annealed (or thermal) andquenched disorder respectively. The connection betweenabsorbing state models in the DP class (without a con-served field) and their interface representation has alsobeen recently considered in the literature [38]. In partic-ular the RFT was mapped into rather unusual interfaceequation, not resembling any known interfacial problem.

The paper is structured as follows: We start in Sec-tion II by presenting the RFT-like Langevin equation forthe recently introduced NCDF class. In III we present aprototypical interface model in the LIM class, in partic-ular the cellular automaton by Leschhorn [17] (see also[18]) and work out a derivation of a Langevin equationfor the activity density ρ, paying particular attention to

the way by which the noise can be found. In SectionIV we proceed conversely: we employ a discrete map-ping of a model with absorbing states in the NDCF classinto a continuous interface representation. We end upwith an interface equation, with several quenched noiseterms that reflect the microscopic rules and the thermalnoise applied in them. We discuss at this point the noisecorrelations that arise and their relevance, with the aidof the renormalization group (RG) literature. Finally, wepresent a discussion and an appendix in which we outlinethe relations between the exponents in the two differentpictures.

II. THE NCDF FIELD THEORY

One particular system in the NDCF class (out of themany studied [13,24]) is a two-species reaction-diffusionmodel, in which one of the species in immobile [11] (seesection IV for a detailed definition). It has the great ad-vantage of allowing for a rigorous derivation of a coarse-grained field theory (or, equivalently, a Langevin equa-tion) via a Fock space representation of the dynamics[32,11,24]. The result is in the form of a Reggeon fieldtheory coupled to an extra conserved non-diffusive field,or what is equivalent, a RFT equation with an extranon-Markovian term [24,5,25]. Quite remarkably thisLangevin equation coincides (up to irrelevant terms) withthe one proposed previously, based only on symmetry andrelevancy arguments, as the minimal Langevin equationcapturing the physics of NDCF, namely [5,24]:

∂ρ(x, t)

∂t= aρ(x) − bρ(x)2 + ∇2ρ(x, t) − µψ(x, t)ρ(x, t)

+σ√

ρ(x, t)η(x, t)

∂ψ(x, t)

∂t= D∇2ρ(x, t), (3)

plus higher order terms, irrelevant from naive powercounting analysis [33]. Note that the second equation, de-scribing the evolution of the background conserved field(coarse grained representation of the total number of par-ticles, which is conserved in the microscopic model), rep-resents an static non-diffusive field: in the absence ofactivity its dynamics is frozen. Observe also that thesecond equation, being linear, can be integrated out, anda closed equation for the activity written down. Moreconcretely

ψ(x, t) = ψ(x, 0) +D

∫ t

0

dt′∇2ρ(x, t′). (4)

The first contribution in Eq(4), a quenched (columnar)disorder, represents the initial condition, while the sec-ond is a non-Markovian term. The Langevin equation(3), even though it looks rather similar to the RFT, hasresisted all renormalization attempts; therefore predic-tions about critical exponents coming from an epsilonexpansion calculation are not available so far.

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III. PHENOMENOLOGICAL ACTIVITY

DESCRIPTION OF LIM MODELS

We consider a representative of the LIM class, namelythe Leschhorn-Tang (LT) cellular automaton [17]. In or-der to study its relation with standard systems with ab-sorbing states, we intend to cast it into a Langevin equa-tion describing the evolution of an activity field [2].

The LT automaton is defined as follows. The inter-face field h(x) satisfies at each discrete time step ti thefollowing equation:

h(x, ti+1) =

{

h(x, ti) + 1, f(x, ti) > 0h(x, ti), f(x, ti) ≤ 0

(5)

where the force f is given by the combination of elasticityand a random quenched pinning force as

f(x, ti) = ∇2h(x, ti) + η(x, h) (6)

where ∇2h(x) is the discrete Laplacian, i. e.∑

nn h(nn)−2Dh(x) where nn denotes the nearest neighbors on ahyper-cubic lattice. A reasonable choice for the noiseis

η(x, h) ={

+1, p−1, 1 − p

(7)

when p is a random number uniformly distributed be-tween zero and unity. This choice implies that the aver-age driving force is F = 〈f〉 = 2p − 1. F plays the roleof a control parameter. The critical point is estimated tobe at pc ∼ 0.800 [17].

At every time step, and at each site where the totaldriving force exceeds its threshold value, i.e., at eachinterface-site advance, we define an activity variable andset it equal to one. On the other hand, in the remaininglattice sites the corresponding activity takes a zero value.Additionally, we also define at each site and time, a con-tinuous “background” variable, equal to ∇2h(x, t) + F .This controls the probability of each interface site to ad-vance at each time, regardless of whether it actually slipsor not. Let us emphasize that this background variableis a conserved magnitude, i.e., it takes a constant value,equal to F when integrated (summed) over the whole lat-tice. However, locally, it favors or inhibits the generationof new activity. We now build up a couple of equationsfor the evolution of the two fields: the activity, ρ(x, t),and the background field, ψ(x, t), which are the coarsegrained field analogous of the previously defined sitevariables. Using the identification between activity and

ready-to-advance sites: h(x, t) =∫ t

0dt′ρ(x, t′) + h(x, 0).

Let us write down a couple of mean-field equations forthe two defined fields:

∂ρ(x, t)

∂t= −ρ(x, t) + ρ(x, t) G [ψ(x, t)]∇2ρ(x, t) (8)

ψ(x, t) ≡ ∇2h(x, t) + F

=

∫ t

0

dt∇2ρ(x, t) + ∇2h(x, 0) + F (9)

The justification of the different terms is as follows:

• The term −ρ(x, t) describes the decay of ac-tive sites, that after the corresponding interface-advance become, in general, non-active. At a coarsegrained level higher order corrections, as −bρ2(x, t)may also appear. In particular, they might play animportant role in order to prevent the activity fromgrowing unboundedly, i.e. in stabilizing the theory.

• +ρ(x, t)G[ψ(x, t)] represents the fact that activityis created in regions where some activity is alreadypresent, and the rate of creation at each point isa function of the local background field, ψ(x, t).Observe that the total contribution of this termwhen integrated over the whole space has to bezero, but locally it fosters or inhibits the creationof further activity. Again, higher order powers ofρ(x, t) might also be included.

• ∇2ρ(x, t) describes the diffusion of activity. Thisterms appears generically for diffusive systems at acoarse grained scale.

• In what respects the ψ(x, t) field, Eq.(9), we havejust written its definition by equating h(x, t) to thenumber of “topplings” (or activity events) at thatpoint in all the preceding history, plus its initialvalue.

Expanding G[ψ(x, t)] in power series, and keepingonly the leading contribution, we are left with a term+λρ(x, t)ψ(x, t) (where λ is a constant) on the r.h.s. ofEq.(8) (observe that the constant term in the Taylor ex-pansion has to be zero as its integral has to be conserved,as argued before). A posteriori, we shall show that theomitted terms, as well as higher order corrections tothe Laplacian term, are irrelevant in what respects largescale, asymptotic, properties.

In order to account for the system fluctuations (com-pletely ignored so far) we now introduce a noise fieldcontribution to Eq.(8). For that, as it is well knownin field theoretical descriptions of systems with absorb-ing states [2,3], a RFT noise term: σ

√

ρ(x, t)η(x, t) isneeded, where σ is a constant and η a Gaussian whitenoise. This just reflects the fact that, as ρ is a local coarsegrained variable its local fluctuations are proportional toits square-root (see [1–3] and references therein). It alsocaptures the physical key ingredient: wherever activityvanishes locally, fluctuations are canceled [2].

Before proceeding further, let us now discuss why thequenched disorder of the microscopic model can be repre-sented by an annealed noise in our description. The keypoint is the observation that in active regions, i.e. wherethe interface advances, a new noise variable is selectedat every time step and, as the interface does not returnto already passed regions, there is no need to store themicroscopic noise history, and the noise can be freshly

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extracted from its probability distribution after every in-terface advance. Therefore, it is rather obvious that indepinned (active) regions, quenched and annealed noisesare fully equivalent. More subtle is the connection of thetwo types of noises in what respect pinned (absorbing)regions. While the annealed noise, η changes in timeeven if there is no activity in a given region, its vari-ations are completely irrelevant as the noise amplitudeappears multiplied by

√ρ = 0. Noise (including its activ-

ity dependent amplitude) at a given spatial point changesonly whenever activity arrives to it, mimicking perfectlywhat happens in the microscopic interface model, wherepinned regions can be depinned only under the presenceof neighboring moving regions. Therefore, the consid-ered time-dependent noise, reproduces properly all theproperties of the original quenched disorder.

All previous considerations lead finally to the followingLangevin equation for the activity field:

∂ρ(x, t)

∂t= [−1 + λF + λ∇2h(x, 0)]ρ(x, t) + ∇2ρ(x, t)

+λρ(x, t)

∫ t

0

dt′∇2ρ(x, t′) + σ√

ρ(x, t)η(x, t) (10)

where we have substituted ψ by its expression com-ing from Eq.(9). In general, the system is expected tolose memory of the initial state for long enough times,therefore the dependence on ∇2h(x, 0) is expected to bewashed out. However, in some cases, as for instance one-dimensional systems, due to the meager phase space, andthe slow relaxation of the initial condition, this might notbe the case [34].

Performing a perturbative, diagrammatic study of theprevious Langevin equation it is easy to see (already atone loop level) that a new non-linearity (vertex), with thesame degree of relevancy as the nonlinear terms alreadypresent in the theory (i.e. the non-local-in-time vertexand the noise one) is perturbatively generated: ρ2(x, t).In fact, this term could have been introduced also at amean field level, as pointed out before, as a stabilizingterm for the activity equation.

Including all the discussed terms into the equation forρ, and integrating the equation for ψ, we finally obtain:

∂ρ(x, t)

∂t= −aρ(x, t) − bρ(x, t)2 + λρ(x, t)

∫ t

0

dt′ρ(x, t′)

+λ∇2h(x, 0) + ∇2ρ(x, t) + σ√

ρ(x, t)η(x, t) (11)

where a = −1 + Fλ and b > 0 are constants. At thispoint, it is a rather straightforward exercise to verifythat no further relevant terms are generated when in-cluding perturbative (diagrammatic) corrections to thebare theory. Therefore, the resulting Langevin equation isidentical to the one proposed for systems with an infinitenumber of absorbing states and an activity field coupledto an static conserved field Eq. (3) [5,13,24].

Summing up, we have mapped a microscopic model be-longing in the LIM class to the Langevin equation char-acterizing the NCDF class. Though our derivation is not

rigorous, we believe it provides strong evidence that infact LIM and NDCF define the same universality class.

IV. MAPPING A REACTION-DIFFUSION

MODEL TO DEPINNING

In this section we proceed conversely to the previousone: starting from a microscopic model in the NDCFclass we map it onto the LIM continuous equation,Eq.(2). To that end we follow a recipe already appliedto many sandpile models exhibiting SOC [27]. Following[24] we consider a two-species reaction-diffusion processon a Ld lattice, with particles of types A and B involved.At each site i, and at each (discrete) time step the fol-lowing reactions take place:

Bi → Bnn , rd ≡ 1 (12)

Ai +Bi → 2 ×Bi , r1 (13)

Bi → Ai , r2. (14)

The Ai, Bi denote particles of each kind at site i. ther’s are the probabilities for the microscopic processes tooccur: diffusion, rd, activation r1, and passivation r2.Without loss of generality we will fix rd = 1, implyingthat, after having the chance to react, B particles dif-fuse with probability one. Thus one can define a phaseboundary between the active and absorbing phases interms of the r1, r2 probabilities, with a phase transitionin-between. We assign occupation numbers nA,i, nB,i toeach site. As the A particles are non-diffusive, this sys-tem has an infinite amount of absorbing states definedby nB,i = 0 for all i, with nA,i arbitrary.

Now we define (analogously to what is done for sand-piles [27,5]) a height field H(x, t) which increases by oneunit every time a site gives one (or more than one) ac-tive, diffusing B particle to one (or more than one) of itsneighbors. When this happens we say, using the sand-pile terminology, that the site “topples”. In this way, theH-field measures the integrated activity at x up to timet.

The mapping to an interface automaton with quenchednoise is based on the fact that both, the reactions betweenthe A and B species, and the diffusion of particles canbe accounted for by looking at their net effects at everytime B particles leave the site x. One just have to look atnA and nB when the site becomes active and a particlediffuses out. The dynamics of H can be written as:

H(x, t+ 1) =

{

H(x, t) + 1 f(x, H) > 0H(x, t), f(x, H) ≤ 0

(15)

which is formally identical to the Leschhorn automatonin the LIM class, with a local “force” defined as

f(x, H) = ntot(x, H) − ξ(x, H) (16)

4

where ntot(x,H) = nA(x, H)+nB(x, H) is the total num-ber of particles at x, and ξ is a local random thresholdthat results from the microscopic processes. More con-cretely, the noise ξ is defined as follows: Consider thesite x after the H-th toppling, either nB(x, H) = 0 ornB(x, H) > 0 (this last can be the case if and only ifparticles have arrived from the nearest neighbors at thesame time step). In the first case, it will remain zero untila particle arrives from a nearest neighbor site; then one isfree to choose a value for ξ(x, H) such that it makes theforce f negative in the time interval between topplings Hand H+1. In the second case, nB(x) will fluctuate owingto the microscopic passivation and activation processes,either going to nB(x) = 0 or inducing a toppling at thenext time step. The relative probabilities of these twoalternatives, as derived from the microscopic dynamics,are captured in the ξ(x, H) probability distribution.

Observe that ξ depends solely upon the total number ofparticles after the preceding toppling and the microscopicdynamical rules. In particular, the larger ntotal the largerthe probability to have many B particles and the largerthe probability to topple. Let us also remark that the im-mobile grains nA constitute a “pinning force” (the largertheir relative number, the lesser the probability to top-ple). The point-wise noise field ξ(x, H) should have weaktwo-point correlations in x since, in particular, it is de-pendent on the number of grains received from the nn’s atthe interface location H(x) which induces weak site-sitecorrelations. The fact that nA changes slowly will makethe H-part of the noise correlator 〈ξ(x, H)ξ(x

′

, H′

)〉 lesstrivial than a simple delta-function δ(H −H ′).

Equation (15) can be considered as a discrete interfaceequation

∆H

∆t= θ (f(x, t)) . (17)

It can be rewritten with the help of two particle-fluxes:ninx

and noutx

, are the number of grains added to or re-moved from a given site x up to time t, respectively.Let us also define g as the average number of particlesgiven to the nearest neighbors at each toppling event. Itis clear that for long enough times nout

x≈ gH(x); rela-

tive deviations from this equality being negligible asymp-totically. Defining also the average value of nin

x, nin

x,

as ninx

= g/2d∑

xnnH(xnn, t), we can compute a noise

τ(x, H) as the deviation of ninx

with respect to its averagevalue:

τ(x, t) = ninx

− g

2d

∑

xnn

H(xnn, t). (18)

In other words, τ counts the relative proportion of parti-cles diffused out from the neighbors that actually arriveto the site under consideration, compared with its av-erage value. A site to which particles have toppled inexcess will take a positive value of τ , and therefore willbe more likely to topple in the following time steps.

Plugging this into Eq.(16), and using that ntot(x, t) =ntot(x, 0) + nin

x− nout

x, we can write [27]

f =g

2d∇2H + F (x, 0) − ξ(x, H) + τ(x, H) (19)

where F (x, 0) ≡ ntot(x, t = 0).

The discretization in Eq.(17) can be understood sothat the rules result in an effective force f ′ that is ex-actly unity when the interface field H advances. Thus∆H/∆t ≡ f ′ θ(f) = f ′θ(f ′) [27]. This construction canbe achieved by picking ξ to have exactly the right valuein order to make the force driving the interface equal tounity, if it is larger than zero. One arrives finally at thediscretized interface equation

∆H

∆t=

g

2d∇2H + F (x, 0) − ξ(x, H) + τ(x, H). (20)

Let us stress the presence of three different noise terms:

1. F (x, 0) represents the original total-number-of-particle configuration at t = 0, and is thereforea columnar noise term [20]. It induces an ini-tial transient regime until eventually, the dynam-ics washes out the dependence of the original con-figuration. In general, columnar disorder is irrel-evant in the renormalization group sense as com-pared to quenched noise; therefore using relevancyarguments, it could be eliminated, at least in highenough dimensions, close or above the critical onedc = 4. Notice that this statement is equivalentto the LIM symmetry, by which static force fieldsF (x, 0) (independent ofH) is completely equivalentto the existence of a non-trivial initial interface pro-file H(x, t = 0). However, in low dimensional sys-tems, and in particular in d = 1, due to the meagerphase space, relaxation times might be huge, andthe time needed to eliminate the dependence on theinitial particle distribution divergently large [34].

2. The noise term ξ(x, H) represents the local thresh-old, determining whether a site with some B parti-cles topples at a given time or, alternatively, theyare transformed into A particles by microscopicprocesses. It captures the in-site microscopic dy-namics, and depends essentially on ntot, and on themicroscopic probabilities. On a nutshell, it sayshow many of the ntot particles are of type A af-ter the microscopic dynamics has operated in thecorresponding time step: if all ntot are of type Athen ξ > ntot, and f < 0; conversely, if any of theparticles is of type B then ξ < ntot and f > 0. Ob-serve that if the diffusion probability was smallerthan unity, then we should substitute ξ(x, H) bya “thermal noise” ξ(x, t), i.e. ξ would change itsvalue after every time step instead of changing onlyafter each toppling: this is due to the fact that ifrd < 1 then a site x including B-type particles couldnot topple at time t (ξ(x, t) below threshold), anddo so at a future time t′ (ξ(x, t′) above threshold).

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This “thermal noise” would generate a rounding offof the transition, but the critical exponents shouldnot be affected by this irrelevant perturbation [14].Therefore, we stick to the simplest case, rd = 1.

3. The noise term τ keeps track of the Brownian mo-tion of particles; i.e. it takes into account the factthat particles are not homogeneously distributedamong the nn, but one of them is picked up ran-domly for each toppling event. It changes slowlysince the effect of the random choices (directions)on the configuration is slow. This is in particulartrue since the noise τ is conserving, as the num-ber of particles is conserved (and as can be seen byintegrating Eq. (18)). A key point is that, analo-gously to what discussed in the preceeding section,the choice to give a particle to a certain neighborcan be taken to be “quenched”, i. e. chosen in ad-vance at t = 0, or “annealed”, i. e. decided on thespot. The correlator of τ can be generically writtenas

〈τ(x′

, H ′)τ(x, H)〉 ∼ f‖(x′ − x)f⊥(H ′ −H). (21)

The (so far unknown) correlators f‖ and f⊥ reflectthe discrete nature of the choices in the dynam-ics. Two microscopic reasons lead immediately tonon-trivial correlations in τ :

(a) The noises τ at the nn’s of site x are correlateddue to an exclusion effect: If a site gives out a dif-fusing B particle to a neighbor, then all the otherneighbors are excluded. The actual coarse-grainednoise correlations are harder to assess, since thefluctuations in the particle flux that τ measuresmake the interface to fluctuate, and thus a sepa-rable noise correlator as Eq. (21) is hard to com-pute. The easiest way to analyze the correlationsamong the different sites is therefore to computethe noise correlator from numerics of the micro-scopic model, using the noise definition Eq. (18).This programme has been pursued for sandpiles[27].

(b) At each site the noise follows the dynamics of arandom walker. In fact, every time a nearest neigh-bor topples, the choice (give the particle to x or to

a different site) makes it so that f⊥ ∼ (H′ −H)1/2

since at every step τ can go “up” or “down” withrespect to the average.

Therefore, reciprocally to what done in the previoussection, we have mapped the reaction-diffusion processinto an interface equation. The dynamics of this inter-face equation follows exactly the history of a reaction dif-fusion process, the details of which are mapped into thequenched noises ξ and τ , and a columnar noise F (x, 0).Let us remark that the existence of a conservation lawhas played a key role in order to obtain a Laplacian inEq.(20).

Finally, using standard renormalization group argu-ments about the relevancy of different operators, wecan eliminate higher order irrelevant terms and noise-correlations, and then we are left with the LIM equationfor point-disorder [14,16] (see also the Appendix).

It must be emphasized, that the mapping works in bothways, it is evident that the noise construction can be in-verted to yield a reaction diffusion process, that corre-sponds to an interface model, assuming that the originalnoise terms have the right correlation and conservationproperties. The interface model Eq.(20) resembles verymuch the one that corresponds to the Manna sandpileautomaton, with the addition of the ξ-noise term whichis more point-disorder like than the τ -term.

Summing up, reciprocally to what was done in the pre-vious section, in this one, we have constructed a mappingbetween a microscopic model in th NCDF class into theLangevin equation for the LIM class.

V. DISCUSSION

We have presented strong theoretical evidence that,rather generically, the universality class of systems withmany absorbing states and order parameter coupled to anon-diffusive conserved field, the NDCF class, and thatof the linear interface model with point-disorder coincide.This fact, already pointed out from numerical simulations[13,24,5] is true at least nearby the critical dimensiondc = 4, where relevancy arguments are reliable. In lowdimensional systems this equivalence could break downowing to the existence, for example, of slow decaying ini-tial conditions [34]. For the frozen configurations in thepoint-disorder LIM it is known that the correlations ofthe forces η(x,H) acting on the interface vanish. In thecase of NDCF models, like the Manna sandpile, such cor-relations (now computed from the particle configurationin frozen configurations) may become non-zero: this isa future avenue for numerical studies, but hopefully thiswould be a irrelevant feature.

Likewise, if one considers a noise field for the LIM(Eq. (7)) with non-trivial (power-law) bare correlationsin x or H , it is unclear at this point how these shouldbe reflected in the construction of a Langevin equationfor the corresponding activity field, like Eq. (10). Corre-lations in the local forces (or “activity thresholds”) willaffect the way the coarse-graining works. For instance,due to the noise structure the pinned and still-active re-gions will be correlated.

In order to establish the connection between the twoclasses we have mapped a discrete interface model intothe Langevin equation characteristic of the NDCF class,and conversely mapped a discrete model in the NDCFinto the well known Langevin equation describing theLIM class. In order to have a more rigorous proof, oneshould be able to map one Langevin equation into the

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other, but this, being the Langevin equations coarse-grained representations of the microscopic models, is notan easy task to fulfill, and remains an open challenge.

Let us remark that a similar problem remains open;namely, the rigorous connection between the QuenchedKPZ [19,6] depinning transition and directed percolationdepinning [7,36] in two dimensional systems, (and to di-rected surfaces in higher dimensions [37]). It is clear fromnumerics, that indeed these two universality classes coin-cide, but a satisfactory proof is, to the best of our knowl-edge, still lacking.

It was the hope, that the possibility of renormalizingthe NDCF Langevin equation using standard RG tech-niques, of problem from the RFT-like equation approach,could shed some light on the (in principle, technicallymore difficult and obscure) functional renormalizationgroup analysis of the interface equation with quenchednoise. However, the difficulties encountered in renormal-izing, using standard perturbative schemes, the Langevinequation for NDCF [24,25] are considerable; and havemade all the attempts to renormalize the theory to fallthrough. Renormalizing the NDCF Langevin equationand relating the derived critical exponents to those ob-tained using functional RG for LIM remains an open andvery challenging problem.

Finally let us also point out that all the discussions pre-sented in this work deal with the “constant force” (in theinterface language) or “fixed energy” (in the absorbing-state terminology) ensemble. They can be easily ex-tended to the “constant force” or “slow driving” ensemble[5,27], in which the system self-organizes into its criticalstate. This point is, however, not essential since all evi-dence points to the fact that if two models belong to thethe universality class, they continue to share the sameset of critical exponents upon changing ensemble.

APPENDIX

The scaling of the phase transition in the absorbing-state representation is characterized by the exponents ν⊥,ν‖, z and β. These describe the correlations in the activ-ity ρ in the spatial and time directions, the developmentof the correlations in time, and the behavior of ρ abovethe critical point, respectively. One has the scaling rela-tion

ρ(∆, L) = L−β/ν⊥R(L1/ν⊥∆) , (22)

where ∆ is the distance to the critical point, and R is ascaling function with R(x) ∼ xβ for large x. For L ≫ξ ∼ ∆−ν⊥ we expect ρa ∼ ∆β (here ξ is the correlationlength). When ∆ = 0 we have that ρa(0, L) ∼ L−β/ν⊥ .For ∆ > 0, by contrast, ρa approaches a stationary value,while for ∆ < 0 it falls off as L−d. These can be used toestablish the numerical values of the exponents.

In the interface representation the relevant exponentsare ν, z as above, with the convention that ν ≡ ν‖. Usu-

ally it is assumed that the dynamics is self-affine, whichimplies that ν⊥ = χν‖ [6,7]. This defines the roughnessexponent χ that characterizes the spatial correlations ofthe interface. If “simple scaling” [35,7] holds, then onehas a unique roughness exponent and we can write forthe interface width w

W 2(t, L) ∼{

t2βW , t≪ t×L2α , t≫ t×,

(23)

using also the early-time exponent βw. If simple scal-ing holds, we have the exponent relation βwz = α [35].If only one timescale is present, the growth exponentis related to the activity time-decay exponent, θ, viaθ + βW = 1 [38].

For point-like disorder the first-loop functional renor-malization group result reads χ = (4 − d)/3, and z =2 − (4 − d)/9 [14]; see the extension to second order in[16]. From these, using the exponent relations, the otherexponents follow. For rather generic bare disorder cor-relators the implication is that the full correlator flowsin the renormalization to this “random field” (or point-disorder) fixed point function, and thus the exponents arethe same. However, numerics in particular in 1D impliesthat the real exponents are different from the one-loopresults. This has recently been explained in terms oftwo-loop corrections, but the traditional interpretationhas been in terms of “anomalous scaling” [17,39], mean-ing that as t→ ∞, the typical height difference betweenneighboring sites increases without limit.

ACKNOWLEDGMENTS

We acknowledge partial support from the Euro-pean Network contract ERBFMRXCT980183, from theAcademy of Finland’s Center of Excellence Program, andDGESIC (Spain) project PB97-0842. We thank R. Dick-man, A. Vespignani, R. Pastor-Satorras, K. B. Lauritsen,and S. Zapperi for useful discussions and long standingpleasant collaborations.

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