C. Maes, F. Redig and E. Saada- The Infinite Volume Limit of Dissipative Abelian Sandpiles

8/3/2019 C. Maes, F. Redig and E. Saada- The Infinite Volume Limit of Dissipative Abelian Sandpiles

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The Infinite Volume Limit of Dissipative

Abelian Sandpiles

C. Maes

F. Redig

E. Saada

22nd October 2003

Abstract: We construct the thermodynamic limit of the stationary measures of theBak-Tang-Wiesenfeld sandpile model with a dissipative toppling matrix (sand grainsmay disappear at each toppling). We prove uniqueness and mixing properties of thismeasure and we obtain an infinite volume ergodic Markov process leaving it invariant.We show how to extend the Dhar formalism of the abelian group of toppling operatorsto infinite volume in order to obtain a compact abelian group with a unique Haar measurerepresenting the uniform distribution over the recurrent configurations that create finiteavalanches1.

1 Introduction

The abelian sandpile is a lattice model where a discrete height-variable (e.g. repre-senting the slope of a sandpile at that site) is associated to each site. (Sand) grainsare randomly added and if at a site the height exceeds some critical value , then thatunstable site topples, i.e., gives an equal portion of its grains to each of its neigh-boring sites which in turn can become unstable and topple etc., until every site hasagain a subcritical height-value. An unstable site thus creates an avalanche involvingpossibly the toppling of many sites around it. The reach of this avalanche depends onthe configuration making this dynamics highly non-local.

Work partially supported by Tournesol project nr. T2001.11/03016VFInstituut voor Theoretische Fysica, K.U.Leuven, Celestijnenlaan 200D, 3001 Leuven, BelgiumFaculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, Postbus 513, 5600 MB

Eindhoven, The Netherlands and EURANDOM, EindhovenCNRS, UMR 6085, Laboratoire de mathematiques Raphael Salem, Universite de Rouen, site Col-

bert, 76821 Mont-Saint-Aignan Cedex, France1MSC 2000: Primary-82C22; secondary-60K35.

Key-words: Sandpile dynamics, Nonlocal interactions, Interacting particle systems, Thermodynamiclimit, Dissipative systems, Decay of correlations and Mass-gap.

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Since their appearance in [1], sandpile models have been studied intensively. Onephysical motivation is related to what was called self-organized criticality. The steadystate typically exhibits power law decay of correlations and of avalanche sizes with anamazing universality of critical exponents found in many computer simulations and ina wide range of natural phenomena. See e.g. [16] for an overview of various models.

The advantage of the abelian sandpile model lies in its rich mathematical structure,first discovered by Dhar, see for instance [4, 5, 8, 14], and also [12] for a mathematicalreview of the main properties of the model in finite volume. The main technical tool inthe analysis is the abelian group of toppling operators which can be identified withthe set of recurrent configurations.

Our aim is to define the model on infinite graphs or better, to understand how theprocess settles down in a stationary regime as the volume increases. The main problemto overcome is the non-locality or extreme sensitivity to boundary-conditions or surfaceeffects of the dynamics which is of course directly related to its physical interest. Wehave constructed in [9] and [10] the infinite volume standard sandpile process on the

one-dimensional lattice and on homogeneous trees. In the present paper we focus on thethermodynamic limit for dissipative models. There, the infinite graph S is a subgraphof the regular lattice Zd and on each site the height has a critical value N =the maximal number of neighbors of a site in S. The finite volume rule now starts asfollows: choose a site x at random from the volume V and add one grain to it. Supposethat x has Nx nearest neighbors and that the new height at x is + 1. Then, it topplesby giving to each of its nearest neighbors one grain and dissipating Nx grains toa sink associated to the volume. We say that the site x is dissipative when > Nxand the model is dissipative when this happens for a considerable fraction of sites.This condition can be rephrased in terms of the simple random walk on S with a sinkassociated to the dissipative sites: the model is dissipative when the Greens function

decays fast enough in the lattice distance, see (2.12) below for a precise formulation.Dissipative abelian sandpile models have appeared in the physics literature in [15, 11]and [3], where it was argued that dissipation removes criticality, that is, correlationfunctions decay exponentially fast uniformly in the volume. From the point of viewof defining the thermodynamic limit, the main simplification of dissipative models isthat there is a stronger control of the non-locality: more precisely, the probabilitythat a site y is influenced by addition on x decays exponentially fast (or at least in asummable way) in the distance between the sites. Hence avalanche clusters are almostsurely finite. As we will see, the avalanche clusters in a dissipative model behave assubcritical percolation clusters, with a characteristic size (in particular they have afinite first moment).

Dissipative models as studied in the present paper teach us little about the originalgoal of sandpile models, i.e., about self-organized criticality. One gains however inproviding a rather complete mathematical analysis. There are various reasons to beinterested in dissipative models. One still obtains a nonlocal dynamics in analogywith the original models but, as we will show, the nonlocality can be better controlledmathematically. Secondly, one can hope to approach the thermodynamic limit of theoriginal critical model again, by letting the dissipation approach to zero. Thirdly, the

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claim in the physics literature that the dissipative model is noncritical has only beenproved for very special explicitly computable correlation functions. We give a completeanalysis and prove that correlations of all local observables decay exponentially. Finally,it will turn out that the dissipative model shows the interesting structure of a compactabelian group of addition operators, also in the infinite volume limit. That generalizes

the Dhar formalism to infinite volume and can stand example of what to expect for theorginal critical model.

1.1 Results

Our three main results are:

1. The extension of the Dhar formalism to infinite volume sandpile dynamics. Thatincludes the construction of a compact abelian group of recurrent configurationson which we can define addition (of sand) operations.

2. The construction of the thermodynamic limit of the finite volume stationary mea-sure with exponential decay of correlations in the case of strong dissipativity.

3. The construction of an infinite volume sandpile process which converges exponen-tially fast to its unique stationary measure.

1.2 Plan of the paper

The paper is organized as follows: in Section 2 we repeat some of the basic results on theabelian sandpile model in finite volume and we introduce the definition of dissipativity,

with examples. In Section 3 we show how to extend the dynamics on infinite volumerecurrent configurations and we recover the group structure of addition of recurrentconfigurations. In Section 4 we prove existence and ergodic properties of the infinitevolume dynamics. Section 5 is devoted to the proof of exponential decay of correlations.

2 Finite volume model

In this section we recall some definitions and properties of abelian sandpiles in finitevolume. In [4], [5], [8], [14] and [12], the reader will find more details.

The infinite graphs S on which we construct the dissipative abelian sandpile dy-namics are S = Zd, and strips, that is, S = Z {1, . . . , }, for some integer > 1(notice that = 1 corresponds to S = Zd with d = 1). Finite subsets of S will bedenoted by V, W; we write S = {W S : W finite}. We denote by V the externalboundary of V: all the sites in S \ V that have a nearest neighbor in V. Let N bethe maximal number of neighbors of a site in S, e.g., N = 2d for S = Zd and N = 4for S = Z {1, . . . , }, 3. The state space of the process in infinite volume is = {1, . . . , }S, with some integer N.

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We fix V S, a nearest neighbor connected subset of S. Then V = {1, . . . , }V is

the state space of the process in the finite volume V. We denote by NV(x) the numberof nearest neighbors of x in V.

A (infinite volume) height configuration is a mapping from S to N = {1, 2,...}assigning to each site x a number of sand grains (x) 1. If , it is called a

stable configuration. Otherwise is unstable. For , V is its restriction to V,and for , , VVc denotes the configuration whose restriction to V (resp. V

c)coincides with V (resp. Vc).

The configuration space is endowed with the product topology, making it into acompact metric space. A function f : R is local if there is a finite W S suchthat W = W implies f() = f(). The minimal (in the sense of set ordering) such Wis called the dependence set of f, and is denoted by Df. A local function can be seenas a function on W for all W Df, and every function on W can be seen as a localfunction on . The set L of all local functions is uniformly dense in the set C() of allcontinuous functions on .

2.1 The dynamics in finite volume

The toppling matrix on S is defined by, for x, y S,

xx = ,

xy = 1 if x and y are nearest neighbors,

xy = 0 otherwise (2.1)

We denote by V the restriction of to V V.

A site x V is called a dissipative site in the volume V ifyV

xy > 0.

Thus if > N, every site is dissipative. If = N, the internal boundary sites of V(that is all the sites in V that have a nearest neighbor in S\ V), are the only dissipativesites in V.

To define the sandpile dynamics, we first introduce the toppling of a site x as themapping Tx : N

V NV defined by

Tx()(y) = (y) Vxy if (x) >

Vxx,

= (y) otherwise. (2.2)

In words, site x topples if and only if its height is strictly larger than Vxx = , bytransferring Vxy {0, 1} grains to site y = x and losing itself in total

Vxx = grains.

As a consequence, if the site is dissipative, then, upon toppling, some grains are lost.Toppling rules commute on unstable configurations, that is, for x, y V such that(x) > = Vxx and (y) > =

Vyy :

Tx (Ty()) = Ty (Tx())

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For NV, we say that V arises from by toppling if there exists a k-tuple(x1, . . . , xk) of sites in V such that

= (k

i=1Txi)()

The toppling transformation is the mapping T : NV V defined by the requirementthat T() arises from by toppling. The fact that stabilization of an unstable configu-ration is always possible follows from the existence of dissipative sites. The fact that Tis well-defined, that is, that the same final stable configuration is obtained irrespectiveof the order of the topplings, is a consequence of the commutation property, see [12] fora complete proof.

For NV and x V, let x denote the configuration obtained from by addingone grain to site x, that is x(y) = (y) + x,y. The addition operator defined by

ax,V : V V; ax,V = T(x) (2.3)

represents the effect of adding a grain to the stable configuration and letting a stableconfiguration arise by toppling. Because T is well-defined, the composition of additionoperators is commutative. We can now define a discrete time Markov chain {n : n 0}on V by picking a point x V randomly at each discrete time step and applying theaddition operator ax,V to the configuration. We define also a continuous time Markovprocess {t : t 0} with infinitesimal generator

L0,V f() =xV

(x)[f(ax,V) f()]; (2.4)

this is a pure jump process on V, where : S (0, ) is the addition rate function.

2.2 Recurrent configurations, invariant measure

The Markov chain {n, n 0} (or its continuous time version {t}) has a uniquerecurrent class RV, and its stationary measure V is the uniform measure on that class,that is,

V =1

|RV|

RV

. (2.5)

A configuration V belongs to RV if it passes the burning algorithm (see [4]),

which is described as follows. Pick V and erase the set E1 of all sites x V with aheight strictly larger than the number of neighbors of that site in V, that is, satisfyingthe inequality

(x) > NV(x)

Iterate this procedure for the new volume V \ E1, and so on. If at the end somenon-empty subset Vf is left, satisfies, for all x Vf,

(x) NVf(x)

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The restriction Vf is then called a forbidden subconfiguration (fsc). IfVf is empty, theconfiguration is called allowed. The set AV of allowed configurations coincides with theset of recurrent configurations, AV = RV (see [8], [12], [14]).

A recurrent configuration is thus nothing but a configuration without forbiddensubconfigurations. This extends to infinite volume:

Definition 2.6 A configuration is called recurrent if for any V S, V RV.

The set R of all recurrent configurations forms a perfect (hence uncountable) subset of. This means that R is closed (hence compact) and every element R is the limitof a sequence n R, n = .

On the set RV, the finite volume addition operators ax,V can be inverted and theygenerate a finite abelian group. This group is characterized by the closure relation

yVaVxyy,V = Id (2.7)

By the group property, the uniform measure V is invariant under the action of ax,Vand ofa1x,V.

2.3 Toppling numbers

For x, y V and V, let nV(x,y,) denote the number of topplings at site y byadding a grain at x, that is, the number of times we have to apply the operator Ty tostabilize x in the volume V. We have the relation

(y) + x,y = ax,V(y) +zV

Vyz nV(x,z,) (2.8)

Defining

GV(x, y) =

V(d) nV(x,y,) (2.9)

one obtains, by integrating (2.8) over V:

GV(x, y) = (V)1xy . (2.10)

In the limit V S, GV converges to the Greens function G of the simple random walk

on S with a sink associated to the dissipative sites (that is every site x is linked with NS(x) edges to a sink and the walk stops when it reaches the sink). By (2.9), theprobability that a site y topples by addition at x in volume V is bounded by GV(x, y).

Definition 2.11 We say that the sandpile model is dissipative if

supxS

yS

G(x, y) < + (2.12)

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In our examples, if > 2d for Zd or 4 for strips, the Greens function G(x, y)decays exponentially in the lattice distance between x and y and hence (2.1) defines adissipative model. From now on, we restrict ourselves to these cases.

Definition 2.13 For any integer n, let Wn be a probability measure on Wn, with

Wn S, Wn S. Then Wn converges to a probability measure on if for anyf L,

limn

f dWn =

fd.

We denote by I the set of all limit points of {V : V S} in the sense of Definition2.13. By compactness of , I is a non-empty compact convex set. Moreover, by (2.5)and Definition 2.6, any I concentrates on R (see [10]).

2.4 Untoppling numbers

On the set RV the addition operators ax,V are invertible. The action of the inverseoperator on a recurrent configuration can be defined recursively as follows, see [8].Consider RV and x V. Remove one grain from at site x. If the resultingconfiguration is recurrent, it is a1x,V, otherwise it contains a forbidden subconfiguration(fsc) in V1 V. In that case untopple the sites in V1. By untoppling of a site z wemean that the sites are updated according to the rule (y) (y) + zy . Iterate thisprocedure until a recurrent configuration is obtained: the latter coincides with a1x,V.As an example, consider a graph with just three sites a b c for = 2. Theconfiguration 212 is recurrent. After removal of one grain at site c, we get 211, whichcontains the fsc 11. Untoppling site b gives 130, and untoppling site c gives 122, which isrecurrent. Conversely, one verifies that addition at site c on 122 gives back the original

configuration 212.Call nV(x,y,) the number of untopplings at site y by removing one grain from x andfrom untoppling sites until a recurrent configuration is obtained. As in the previoussection, one easily proves the relation

nV(x,y,)V(d) = GV(x, y) (2.14)

3 The group of addition operators in infinite volume

In this section we show how to obtain the group of addition operators in the infinitevolume limit. The assumption of dissipativity is crucial in order to obtain a compactabelian group in the thermodynamic limit.

3.1 Addition operator

The finite volume addition operators ax,V (cf. (2.3)) are defined on via

ax,V : : ax,V = (ax,VV)VVc. (3.1)

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(with some slight abuse of notation). Similarly, the inverses are defined on R via

a1x,V : R : (a1x,VV)VVc (3.2)

Remark that if R, then (ax,V)W RW for all W V but ax,V is not necessarilyan element of R.

Definition 3.3 For , we say that the limit of the finite volume addition operatorsis defined on if for every x S, there exists 0 S such that for any S, 0,ax, = ax,0; in that case, we write

ax = ax,0

Similarly, for R, we say that the limit of the finite volume inverse addition operatorsis defined on if for every x S, there exists 0 S such that for any S, 0,a1x, = a

1x,0

; we write

a1x = a1x,0

Remark that if R and ax is defined on , then ax R.

Lemma 3.4 Assume (2.12). For any I there exists a tail measurable subset such that:

1. () = 1;

2. The limit of the finite volume addition operators and their inverses is defined onevery .

Moreover, every I is invariant under the action of ax and a1x , that is, for all

x S and f L f(ax)(d) =

f(a1x )(d) =

f()(d) (3.5)

and axa1x = a

1x ax = id on .

Proof. We prove the result for the addition operators, the analogue for the inverses isproved along the same lines by replacing number of topplings by number of untop-plings.

Pick Wk S, Wk S such that Wk and x S. We have to prove that

[0 S, V 0 : ax,V = ax,0] = 0 (3.6)

We enumerate S = {xn : n N}, with Vn = {x1, . . . , xn} such that Vn S, xn Vn1.If ax,V = ax,Vn, then some boundary site of Vn has toppled under addition at x involume V. This implies that for every m such that Vm V some external boundarysite of Vn topples upon addition at x in Vm. Therefore, the left hand side of (3.6) isbounded by

n N, p n, y Vn : nVp(x,y,) 1

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and we have to estimate

p n, y Vn : nVp(x,y,) 1

(3.7)

Since nVp(x,y,) nVp+1(x,y,),

p n, y Vn : nVp(x,y,) 1 limk

(y Vn : nVk(x,y,) 1)

limk

yVn

nVk(x,y,)(d)

limk

yVn

nWk(x,y,)Wk(d)

=

yVn

G(x, y)

which implies that (3.7) converges to zero as n tends to infinity, by condition (2.12).Finally, (3.5) follows easily from Definition 3.3, () = 1, f L, and the invariance ofV under the finite volume addition operators ax,V and a

1x,V.

Notice that for , we can take the limit V S in (2.8) and write

(y) + x,y = ax(y) +zS

yz nS(x,z,) (3.8)

for any x, y S, where nS(x,z,), the number of topplings at site z S by adding agrain at x, satisfies zS nS(x,z,) < +.Lemma 3.9 Assume (2.12). For any I there exists a tail measurable subset o with(o) = 1 such that for anyV S and nx, x V integers, the product

xV a

nxx

is well-defined, as the limit of

xV anxx, as S, on every

o.

Proof. We fix V S, x V, nx a positive integer, and we prove that anxx is well-defined on (the case of negative nx is similar and the extension to finite products itstraightforward). Following the same lines as in the preceding proof, we have to replace(3.6) by

0 S, 0 : a

nxx, = a

nxx,0

= 0

We denote by EVp(nx, x , z , ) the event that addition in Vp of nx grains at x causes atleast one toppling at z. As these events are increasing in p, we estimate

p n, y Vn : EVp(nx,x,y,) 1

limk

yVn

Wk(EWk(nx,x,y,))

yVn

nxG(x, y)

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where the last inequality is a consequence of (2.10) and (3.5). From this we deduce thatfor any V S, n = (nx, x V) ZV, the product

xV a

nxx is well-defined on a tail

measurable set (V, n) of -measure one. The set o is then the countable intersection

o = VS,n Z V(V, n)

of tail measurable -measure one sets.

The following proposition extends this to addition on infinite products.

Proposition 3.10 Assume (2.12). Ifn = (nx, x S) ZS satisfies

xS |nx|G(0, x) 0 so that for all f L, there exists Cf < + such thatS(t)f d f d Cf exp(C2t) (4.9)In particular, S(t) converges weakly to , uniformly in and exponentially fast.

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Proof. The idea is to approximate S(t) by finite volume semigroups, and to estimatethe speed of convergence as a function of the volume. More precisely, we split

S(t)f d

f d

AVt (f) + BVt (f) + CV(f) (4.10)

with

AVt (f) =

S(t)f d SV(t)fdVBVt (f) =

SV(t)f dV f dVCV(f) =

fdV fd .where V is the restriction of to V and

SV(t)f() = fxV

aNt,x

x,V dPBy Theorem 3.25,

limVS

CV(f) = 0. (4.11)

For the first term in the right-hand side of (4.10) we write

AVt (f) =

f

xS

aNt,x

x

f

xV

aNt,x

x,V

dPd

(4.12)

The integrand of the right hand side is zero if no avalanche from Vc has influenced sitesofDf during the interval [0, t], otherwise it is bounded by 2f. Therefore, since Nt,x

are rate one Poisson processes:

AVt (f) ft

yDf

xVc

G(x, y) (4.13)

for some constant . Therefore that first term can be controlled by the dissipativitycondition (2.12).The second term in the right hand side of (4.10) is estimated by the relaxation toequilibrium of the finite volume dynamics. The generator L0V has the eigenvalues

(L0V) =

xV

exp

2i

yV

GV(x, y)ny

1

: n ZV/VZV

(4.14)

The eigenvalue 0 corresponding to the stationary state arises from the choice n = 0.For the speed of relaxation to equilibrium we are interested in the minimum absolutevalue of the real part of the non-zero eigenvalues. More precisely:

BVt (f) Cf exp(Vt)

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where

V = inf

|Re()| : (L0V) \ {0}

= 2 inf xV sin2

yV GV(x, y)ny : n ZV/VZV, n = 0

by (4.14). Since there is a constant c so that for all real numbers r

sin2(r) c(min{|r k| : k Z})2

we getxV

sin2

((V)1n)x

c inf

(V)1n k2 : n ZV/VZV, n = 0, k ZV

(4.15)where represents the Euclidian norm in ZV that we estimate by

(V)1n k2 = (V)1(n Vk)2 V2

For any regular volume we have

V

22 + 16d2

This givesBVt (f) Cf exp(Ct) (4.16)

where C > 0 is independent of V.The statement of the theorem now follows by combining (4.11), (4.13), (4.16).

Remark 4.17 When we restrict ourselves to the case wherexS

(x) = M < , (4.18)

L becomes a bounded operator, hence it generates a pure jump process which is acontinuous time random walk on the group (R/ , ). By the ergodic properties ofrandom walks on compact groups we then obtain that

limt S(t) = .

for every measure on R/ (see Theorems 2.5.14, 2.6.2 and Corollary 2.6.4 in [7]for details).

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4.3 Mixing property

To the stationary process defined in Theorem 4.4, we associate the process on R/ byputting

[]t = [t]. (4.19)

For that, it is important to notice that the equivalence of recurrent configurations ispreserved in time (by Theorem 4.5, and points 1,2 of Remark 3.18): when ,then t t with P-probability one. For the following theorem, we abbreviate withoutconsequences []t = t

.

Theorem 4.20 The process {t : t 0} is mixing, that is, for all f, g L:

limt

(S(t)f) gd =

f d

gd (4.21)

Proof. Since the semigroup is a normal operator on L2(), ergodicity of the processimplies the mixing property by [13]. It thus suffices to prove that for a bounded non-negative function f such that

fd > 0 and Lf = 0, then -a.s. f =

f d. By the

invariance of under ax,

0 = 2

(f Lf) d =

xS

(axf f)

2d

which implies axf = f for all ax, -a.s. Hence, the measure

df =f df d

is invariant under the action of ax, thus under the group action. By uniqueness of theHaar measure, we conclude f = .

5 Decay of correlations

In this section we prove that the infinite volume measure has exponential decay ofcorrelations under a condition of strong dissipativity. That means for the model (2.1)

with S = Zd that must be sufficiently large, e.g. > 13 for d = 2; for the stripsS = Z {1, . . . , } with finite it always suffices that > 3.In [11] the exponential decay between very special local observables (indicators of so-called weakly allowed clusters) is also obtained in the case where the Green functiondecays exponentially. However, the technique developed in that paper does not applyto all local functions.

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5.1 Decoupling argument

We start with the heuristics of the main ingredient in the proof of exponential decay ofcorrelations. The rest is based on quite general stochastic-geometric methods that arereviewed in [6].

To be specific, suppose that S = Z2

and > 4 (in (2.1)). Then, for every volumeV S,V((x) = a|(z) = c) = V\z((x) = a) (5.1)

for all a, c {1, . . . , }, c > 4, x = z in V, because, by the burning algorithm, wecan burn away the sites on which we know that the configuration is sufficiently large.Instead of fixing in (5.1) the height value at one site z, we could do the same thing onsome region C V that does not contain x, see Lemma 5.4. On the other hand, if sitesx and y are not very close to each other, we can find volumes x, y V that contain x,respectively y, that also do not touch (more precisely, that satisfy (x x) y = ).Then, see Lemma 5.7,

xy((x) = a, (y) = b) = x((x) = a)y((y) = b) (5.2)

The combination of (5.1) with (5.2) yields conditional independence of two events thatare separated by a region C where the configuration is sufficiently high, see Lemma 5.9.

Definition 5.3 LetV S, C V, V. The subconfiguration C is V-burnable ifthere exists a bijection f : {1, . . . , n} C such that

NV(f(1)) < (f(1)),

and for every j = 1, . . . , n 1,

NV\{f(1),...,f(j)}(f(j + 1)) < (f(j + 1)).

As an example, on Z2 with maximal height = xx = 5, every closed curve alongwhich the heights are at least 4 and containing at least one point with height 5 isburnable.

Lemma 5.4 LetV = C S, C = and fix an arbitrary configuration Vso that C is V-burnable. Put

EC = { V : C = C}

Then, for all events A that depend only on the configuration in (that is, A F),

V(A|EC) = (A) (5.5)

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Proof. By the burning algorithm, RV EC if and only if R and C = C.Therefore,

V(A|EC) =

RV

I( A)I( EC)

RVI( EC)

= RCRC I( A)I( EC)CRC

I( EC)|R|

= (A).

Remark 5.6 We do not need to condition on one fixed burnable configuration. TheLemma and its proof above remain unchanged when taking

EC = { V : C is V-burnable}

the event that we can burn away the sites of C first.

Lemma 5.7 Let1, 2 S with

(1 1) 2 = . (5.8)

For A F1 , B F2,

12(A B) = 12(A)12(B)

Proof. We have R12 if and only if1 R1 and 2 R2 . The rest is writingout expectations as in the proof of Lemma 5.4.

We now state the conditional independence.

Lemma 5.9 ForV S andC V, suppose thatV\C = 12 with1, 2 satisfying(5.8). Then, for all A F1, B F2

V(A B|EC) = V(A|EC)V(B|EC)

Proof. By Lemma 5.4,V(A B|EC) = V\C(A B)

and continuing via Lemma 5.7

V(A B|EC) = 12(A)12(B). (5.10)

The proof is finished by applying again Lemma 5.4 to the two factors in the right-hand

side of (5.10).The conditional independence (5.10) is reminiscent of the situation for Markov ran-

dom fields. Here V is not Markovian but nevertheless for all A F, V theconditional probability of A given the configuration in V \ is

V(A|V\) = (A) (5.11)

whenever V is V-burnable. In particular, this conditional probability (5.11) is thenindependent of the particular V\.

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5.2 Geometric argument

From the previous decoupling argument, it is clear how to proceed for the proof ofdecay of correlations. What needs to be established is that there will be typicallysome circuit C, separating two far away dependence sets, where the configuration

is burnable. We thus basically end up with a stochastic-geometric or percolation-likeargument as also reviewed in [6]. The first thing to see is that burnability is sufficientlyprobable. We do that first for the strip in Lemma 5.12 and then for the full lattice inLemma 5.16.

Lemma 5.12 Let V = {(x, y) Z2 : |x| k, y = 1, . . . } and 4 in (2.1). Fixsome x1, |x1| k and let C = {(x1, y) V : y = 1, . . . }. There is p = p(, ) > 0(uniformly in k) such that for all events E(x1) that only depend on the heights (x, y)with (x, y) C,

V((x, y) 4 for all (x, y) C|E(x1)) p > 0 (5.13)

Proof. Via Bayesrule,

V((x, y) 4 for all (x, y) C|E(x1)) = (5.14)

V(E(x1)|(x, y) 4 for all (x, y) C)V((x, y) 4 for all (x, y) C)

V(E(x1))

If (x, y) 4 for the points (x, y) C, then C is V-burnable, and by Lemma 5.4

V(E(x1)|(x1, y1) 4 whenever |y1| ) = V\C(E(x1))

On the other hand, by counting,

V\C(E(x1))

V(E(x1))

|RV|

|RV\C||RC|

and

V((x, y) 4 for all (x, y) C) = ( 4 + 1)

|RV\C|

|RV|

As a consequence we can take

0 < p

( 4 + 1)

|RC| =

3

3 +

Remark 5.15 Obviously, p 0 as .

For the regular lattice S = Zd we have:

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Lemma 5.16 Consider the model (2.1) withS = Zd. For all > 0, there is a0 < +so that for all V S, all x V and all events E FV\x,

V((x) > 2d|E) > 1

whenever 0.

Proof. We can repeat the steps of the proof in Lemma 5.12. At the end we mustestimate the number of burnable heights at x divided by the number of configurationsat x. That is,

V((x) > 2d|E) > 2d

It thus suffices that 2d < .

We need one more lemma before giving the geometric argument, because the latterrequires stochastic domination by Bernoulli measure.

Lemma 5.17 The invariant probability measure V for the sandpile dynamics in Vis irreducible,that is, for two given recurrent configurations , , there is a sequence0 = , . . . , m =

of recurrent configurations such that i and i+1 differ only in onesite.

Proof. Since can be reached from by a finite number of additions, it is enough toshow that for any x V, there is such a sequence from to ax. Let

+x () = {y V; (ax)(y) > (y)},

x () = {y V; (ax)(y) < (y)}.

We first add sand grains one by one on the sites y +x (), to reach the value (ax)(y).Each step leads to a configuration larger than , thus recurrent. We denote by + = +0the recurrent configuration (ax)+x ()[+x ()]c . If

x () = we are finished. If not,

we write x () = {z1, . . . , zn} and we pass from +k = (ax){z1,...,zk}

+{z1,...,zk}c

to +k+1for every 1 k n (+k ax is recurrent and differs from

+k1 in one site) to reach

+n = ax.

We are now in a position to give the main stochastic-geometric argument leadingto exponential decay of correlations. It copies the proof that complete analyticity for

Markovian random fields follows from absence of disagreement percolation, as done in[2], see Theorem 7.1 in [6], except that we can replace the Markov property by thedecoupling property (5.11).

Let pc(d) denote the percolation threshold for Bernoulli site percolation on Zd. Let

V S and let V on which we fix two arbitrary height configurations and toconsider two conditional probabilities 1 = V(|) and 2 = V(|

).

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Theorem 5.18 Suppose that 4 for S = Z {1, . . . , } or 4d < pc(d) for S = Zd

in (2.1). There exist constants > 0, C < + so that for all V S, V, W V \ , R and for every event A FW,

|1(A) 2(A)| Cedist (W,) (5.19)

where dist(, ) is the nearest neighbor distance between the two subsets.

Proof. We give the proof for the lattice S = Zd. The case of the strip is analogous buta little simpler (using Lemma 5.12).We use a coupling argument. First, we introduce some linear ordering on V \ . Weconstruct via iteration a coupling between 1 and 2 which is a random field (X, X

). Westart by setting X(x) = (x), X(x) = (x) on . Suppose that we have already realizedthe coupling as (X, X) = (, ) on all sites outside some non-empty set T V \ .Consider then the conditional distributions V(|V\T) and V(|

V\T). One possibility

is that on the external boundary both T and T are V-burnable. But then, via

(5.11), these two conditional probabilities are equal on T and we can take the optimalcoupling for which X = X on T. Alternatively, we choose the smallest site x Thaving a nearest neighbor y V \ T, for which X(y) 2d or X(y) 2d and wefind the value ((x), (x)) for the coupling at x from sampling the optimal couplingbetween the single site distributions V(X(x) = |V\T) and V(X

(x) = |V\T). At

this step the coupling is defined outside T \ x and we can repeat the iteration giving usa coupling between 1 and 2.From the above construction, it is possible that in the coupling X(x) = X(x) at somex W, only if there is a nearest neighbor path from x to on which X(y) 2d orX(y) 2d. On the other hand, no matter what we fix off y,

P(X(y) 2d or X(y) 2d|(z), (z), z V \ y) 2(1 V((y) > 2d|(z), z = y))(5.20)

For large enough (from Lemma 5.16), this is bounded by pc(d). The proof is then con-cluded via an application of stochastic domination with the Bernoulli product measure(thanks to Lemma 5.17, see Theorem 4.8 in [6]) and using that the cluster-diameter insub-critical Bernoulli site percolation has an exponential tail.

Examples.

1. The dissipative system in dimension 2: we have pc(2) = 0.5927 (as numericalresult). Thus we need to take > 13 so that 8 < pc(2).

2. The dissipative system in high dimension. Since pc(d) 1/(2d) for large d, weconclude exponential decay of correlations as soon as > 8d2.

References

[1] Bak, P., Tang, K. and Wiesenfeld, K., Self-Organized Criticality, Phys. Rev. A 38,364374 (1988).

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[2] van den Berg, J. and Maes, C., Disagreement percolation in the study of Markovfields, Ann. Probab. 25, 13161333 (1994).

[3] Daerden, F., Vanderzande, C., Dissipative abelian sandpiles and random walks,Phys. Rev E 63 (2001).

[4] Dhar, D., Self Organised Critical State of Sandpile Automaton Models, Phys. Rev.Lett. 64, No.14, 16131616 (1990).

[5] Dhar, D., The Abelian Sandpiles and Related Models, Physica A 263, 425 (1999).

[6] Georgii, H.-O., Haggstrom, O., Maes, C., The random geometry of equilibriumphases, Phase Transitions and Critical Phenomena, Vol. 18, Eds C. Domb andJ.L. Lebowitz (Academic Press, London), 1142 (2001).

[7] Heyer, H, Probability measures on locally compact groups, Springer, 1977.

[8] E.V. Ivashkevich, Priezzhev, V.B., Introduction to the sandpile model, Physica A254, 97116 (1998).

[9] Maes, C., Redig, F., Saada E. and Van Moffaert, A., On the thermodynamic limit for a one-dimensional sandpile process, Markov Proc. Rel. Fields, 6, 122 (2000).

[10] Maes, C., Redig, F., Saada E., The abelian sandpile model on an infinite tree, Ann.Probab. 30, No. 4, 127(2002).

[11] Mahieu, S., Ruelle, P., Scaling fields in the two-dimensional abelian sandpile model,Phys. Rev E 64 066130-(1-19) (2001).

[12] Meester, R., Redig, F. and Znamenski, D., The abelian sandpile; a mathematicalintroduction, Markov Proc. Rel. Fields, 7, 509523 (2002).

[13] Rosenblatt, M., Transition probability operators, Proc. Fifth Berkeley Symposium,Math. Statist. Prob., 2, 473483 (1967).

[14] Speer, E., Asymmetric Abelian Sandpile Models, J. Stat. Phys. 71, 6174 (1993).

[15] Tsuchiya, V.T and Katori, M., Phys. Rev E 61, 1183 (2000).

[16] Turcotte, D.L., Self-Organized Criticality, Rep. Prog. Phys. 62, 13771429 (1999).

C. Maes, F. Redig and E. Saada- The Infinite Volume Limit of Dissipative Abelian Sandpiles

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