Christian Maes, Frank Redig and Ellen Saada- The Abelian Sandpile Model on an Infinite Tree

8/3/2019 Christian Maes, Frank Redig and Ellen Saada- The Abelian Sandpile Model on an Infinite Tree

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arXiv:m

ath-ph/0101005v1

5Jan2001

The Abelian Sandpile Model on an

Infinite Tree

Christian Maes, K.U.Leuven

Frank Redig, T.U. Eindhoven

Ellen Saada, C.N.R.S., Rouen

February 7, 2008

Abstract: We consider the standard Abelian sandpile process on the Bethe

lattice. We show the existence of the thermodynamic limit for the finite volume

stationary measures and the existence of a unique infinite volume Markov process

exhibiting features of self-organized criticality1.

1 Introduction

Global Markov processes for spatially extended systems have been around forabout 30 years now and interacting particle systems have become a branch ofprobability theory with an increasing number of connections with the naturaland human sciences. While standard techniques and general results have been

collected in a number of books such as [Liggett (1985), Chen (1992), Toom (1990)]and are capable to treat the infinite volume construction for stochastic systemswith locally interacting components, some of the most elementary questions forlong range and nonlocal dynamics have remained wide open. We have in mindthe class of stochastic interacting systems that during the last decade have in-vaded the soft condensed matter literature and are sometimes placed under thecommon denominator of self-organizing systems.

Since the appearance of the paper [BTW (1988)], the concept of self-organizedcriticality (SOC) has suscited much interest, and is applied in a great variety ofdomains (see e.g. [Turcotte (1999)] for an overview). From the mathematicalpoint of view, the situation is however quite unsatisfactory. The models ex-hibiting SOC are in general very boundary condition dependent (especially theBTW model in dimension 2), which suggests that the definition of an infinite

volume dynamics poses a serious problem. Even the existence of a (unique)thermodynamic limit of the finite volume stationary measure is not clear. From

1MSC 2000: Primary-82C22; secondary-60K35.

Key-words: Sandpile dynamics, Nonlocal interactions, Interacting particle systems, Ther-

modynamic limit.

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Abelian Sandpile Model 2

the point of view of interacting particle systems no standard theorems are atour disposal. The infinite volume processes we are looking for will be non-Feller

and cannot be constructed by monotonicity arguments as in the case of theone-dimensional BTW model (see [MRSV (2000)]) or the long-range exclusionprocess (see [Liggett (1980)]). On the other hand in order to make mathemati-cally exact statements about SOC, it is necessary to have some kind of infinitevolume limit, both for statics and for dynamics.

In this paper we continue our study of the BTW-model for the case of theBethe lattice, this is the abelian sandpile model on an infinite tree. For thissystem, many exact results were obtained in [DM (1990b)]. In contrast to theone-dimensional case this system has a non-trivial stationary measure. We showhere that the finite volume stationary measures converge to a unique measure which is not Dirac and exhibits all the properties of a SOC-state. We then turnto the construction of a stationary Markov process starting from this measure. The main difficulty to overcome is the strong non-locality: Adding a grain atsome lattice site x can influence the configuration far from x. In fact the clusterof sites influenced by adding at some fixed site has to be thought of as a criticalpercolation cluster which is almost surely finite but not of integrable size. Theprocess we construct is intuitively described as follows: At each site x of theBethe lattice we have an exponential clock which rings at rate (x). At theringing of the clock we add a grain at x. Depending on the addition rate (x),we show existence of a stationary Markov process which corresponds to thisdescription. We also extend this stationary dynamics to initial configurationswhich are typical for a measure that is stochastically below .

The paper is organized as follows. In section 2 we introduce standard resultson finite volume abelian sandpile models, and summarize some specific resultsof [DM (1990b)] for the Bethe lattice which we need for the infinite volume

construction. In section 3 we present the results on the thermodynamic limitof the finite volume stationary measures and on the existence of infinite vol-ume Markovian dynamics. Section 4 is devoted to proofs and contains someadditional remarks.

2 Finite Volume Abelian Sandpiles

In this section we collect some results on abelian sandpiles on finite graphs whichwe will need later on. Most of these results are contained in the review paper[Dhar (1999)], or in [IP (1998)].

2.1 Toppling Matrix

Let V denote a finite set of sites and V = (Vx,y)x,yV a matrix indexed bythe elements of V satisfying the conditions :

1. For all x, y V, x = y, Vx,y = Vy,x 0,

2. For all x V, Vx,x 1,

3. For all x V,

yV Vx,y 0,

4.

x,yV Vx,y > 0.


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Such a matrix V is called a toppling matrix. The fourth condition ensures thatthere are sites (so called dissipative sites) for which the inequality in the third

condition is strict. This is fundamental for having a well defined toppling rulelater on. In the rest of the paper we will choose V to be the lattice Laplacianwith open boundary conditions on a finite simply-connected set V S, whereS is a regular graph, like the d-dimensional lattice d, or the infinite rootlesstree ITd of degree d + 1. More explicitly:

Vx,x = 2d if V d,

= d + 1 if V ITd,

Vx,y = 1 if x and y are nearest neighbors. (2.1)

The dissipative sites then correspond to the boundary sites of V.

2.2 ConfigurationsA height configuration is a mapping from V to IN = {1, 2,...} assigning toeach site a natural number (x) 1 (the number of sand grains at site x). Aconfiguration INV is called stable if, for all x V, (x) Vx,x. Otherwise is unstable. We denote by V the set of all stable height configurations. For INV and V V, |V denotes the restriction of to V.

2.3 Toppling Rule

The toppling rule corresponding to the toppling matrix V is the mapping

TV : INV V INV

defined by

TV (, x)(y) = (y) Vx,y if (x) >

Vx,x,

= (y) otherwise. (2.2)

In words, site x topples if and only if its height is strictly larger than Vx,x, by

transferring Vx,y grains to site y = x and losing itself Vx,x grains. Toppling

rules commute on unstable configurations. This means for x, z V and suchthat (x) > Vx,x, (z) >

Vz,z ,

TV (TV (, x), z) = TV (TV (, z), x) (2.3)

We write [TV (, z)TV (, x)]() = [TV (, x)TV (, z)]().

Choose some enumeration {x1, , xn} of the set V. The toppling transfor-mation is the mapping

TV : INV V

defined by

TV () = limN

ni=1

TV (, xi)

N(). (2.4)

In [IP (1998)] it is recalled that


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1. The limit in (2.4) exists, i.e. there are no infinite cycles, due to the presenceof dissipative sites.

2. The stable configuration TV () is independent of the chosen enumerationof V. This is the abelian property and follows from (2.3).

2.4 Addition Operators

For INV and x V, let x denote the configuration obtained from byadding one grain to site x, i.e. x(y) = (y) + x,y. The addition operatordefined by

ax,V : V V ; ax,V = TV (x) (2.5)

represents the effect of adding a grain to the stable configuration and lettingthe system topple until a new stable configuration is obtained. By (2.3), thecomposition of addition operators is commutative: For all V , x, y V,

ax,V(ay,V) = ay,V(ax,V).

2.5 Finite Volume Dynamics

Let p denote a non-degenerate probability measure on V, i.e. numbers px,0 < px < 1 with

xV px = 1. We define a discrete time Markov chain

{n : n 0} on V by picking a point x V according to p at each discretetime step and applying the addition operator ax,V to the configuration. ThisMarkov chain has the transition operator

PVf() =xV

pxf(ax,V). (2.6)

We can equally define a continuous time Markov process {t : t 0} withinfinitesimal generator

LVf() =xV

(x)[f(ax,V) f()], (2.7)

generating a pure jump process on V, with addition rate (x) > 0 at site x.

2.6 Recurrent Configurations, Stationary Measure

We see here that the Markov chain {n, n 0} has only one recurrent class andits stationary measure is the uniform measure on that class.

Let us call RV the set of recurrent configurations for {n, n 0}, i.e. those

for which P(n = infinitely often) = 1, where P denotes the distribution of{n, n 0} starting from 0 = V. In the following proposition we listsome properties of RV. For the sake of completeness we include a proof whichwe could not find worked out completely in the literature.

Proposition 2.1

1. RV contains only one recurrent class.

2. The composition of the addition operators ax,V restricted to RV definesan abelian group G.


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3. |G| = |RV|.

4. For anyx V, there exists nx such that for any RV, anxx,V = .

5. |RV| = det V.

Proof: 1. We write if in the Markov chain can be reached from withpositive probability. Since sand is added with positive probability on all sites(px > 0), the maximal configuration max defined by

max(x) = Vx,x

can be reached from any other configuration. Hence, if RV then max,therefore max RV and max (see e.g. [Chung (1960)] p.19).

2. Fix RV ; then there exist ny 1 such that

yV

anyy,V = ,

andgx = a

nx1x,V

yV,y=x

anyy,V

satisfies (ax,Vgx)() = (gxax,V)() = . The set

Rx = { RV : (ax,Vgx)() = }

is closed under the action of ax,V, contains , hence also max: it is a recurrentclass. By part 1, Rx = RV , ax,Vgx is the neutral element e, and gx = a

1x,V if

we restrict ax,V to RV.3. Fix RV and put : G RV; g g(). As before (G) is a

recurrent class, hence (G) = RV . If for g, h G, (g) = (h), then

gh1() = , and by commutativity gh1(g) = g for any g G. Thereforegh1() = for all RV, thus g = h. This proves that is a bijection fromG to RV.

4. Since G is a finite group, for any x V there exists nx 1 such thatanxx,V = e.

5. Adding Vx,x particles at a site x V makes the site topple, and Vx,y

particles are transferred to y. This gives

aVx,xx,V =

y=x

aVx,yy,V .

On RV the ax,V can be inverted and we obtain the closure relation

yV

aVx,y

y,V= e,

which completely determines the one-dimensional representations of the groupof addition operators, and in particular the cardinality of the latter, as obtainedin [Dhar (1990a)].

Remark. RV does not depend on the px, and does not change by going fromdiscrete to continuous time, i.e. from (2.6) to (2.7).

The main consequence of the group property of G is the fact that the uniquestationary measure is uniform on RV .


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Proposition 2.2

1. The measureV =

RV

1

|RV | (2.8)

is invariant under the action ofax,V, x V ( is the Dirac measure onconfiguration ).

2. On L2(V) the adjoint of ax,V is

ax,V = a1x,V. (2.9)

Proof: Since ax,V : RV RV can be inverted, we have

RV

f(ax,V)g() = RV

f()g(a1

x,V

),

hence (2.9). By choosing g 1, part 1 follows.

Remark. This shows that V is invariant under the Markov processes gener-ated by (2.6) and (2.7).

2.7 Burning Algorithm

The burning algorithm determines whether a stable configuration V isrecurrent or not. It is described as follows: Pick V and erase all sitesx V satisfying the inequality

(x) >

yV,y=x

(Vx,y).

This means erase the set E1 of all sites x V with a height strictly larger thanthe number of neighbors of that site in V. Iterate this procedure for the newvolume V \ E1, and the new matrix V\E1 defined by

V\E1x,y = Vx,y if x, y V \ E1

= 0 otherwise,

and so on. If at the end some non-empty subset Vf is left, satisfies, for allx Vf,

(x) yVf ,y=x

(Vx,y).

The restriction |Vf is called a forbidden subconfiguration. If Vf is empty, theconfiguration is called allowed, and the set AV of allowed configurations satisfies

Proposition 2.3

AV = RV.


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The main ingredient to prove this result (see [IP (1998)], [Speer (1993)]) isthe fact that toppling or adding cannot create a forbidden subconfiguration.

The set AV is thus closed under the dynamics and contains the maximal con-figuration max.

Remark that the burning algorithm implies that for V V, and V, RV, then any V such that |V = satisfies RV . Indeed,the property of having a forbidden subconfiguration in Vf only depends onthe heights at sites x Vf. Therefore RV implies |V RV. Thisconsistency property will enable us to define allowed configurations on infinitesets.

2.8 Expected Toppling Numbers

For x, y V and V, let nV(x, y, ) denote the number of topplings at sitey V by adding a grain at x V, i.e. the number of times we have to applythe operator TV (, y) to relax x. Define

GV(x, y) =

V(d) nV(x, y, ). (2.10)

Writing down balance between inflow and outflow at site y, one obtains (cf.[Dhar (1990a)])

zV

Vx,zGV(z, y) = x,y,

which yieldsGV(x, y) = (

V)1x,y.

In the limit V S (where S is d or the infinite tree), GV converges to the

Greens function of the simple random walk on S.

2.9 Some specific results for the tree

When Vn is a binary tree of n generations, many explicit results have beenobtained in [DM (1990b)]. We summarize here the results we need for theconstruction in infinite volume.

1. When adding a grain on a particular site 0 Vn of height 3, the set oftoppled sites is the connected cluster C3(0, ) of sites including 0 havingheight 3. This cluster is distributed as a random animal (i.e. its distribu-tion only depends on its cardinality, not on its form). Moreover

limnVn (|C3(0, )| = k) Ck3/2

(2.11)

as k goes to infinity. The notation means that if we multiply the lefthand side of (2.11) by k3/2, then the limit k is some strictly positiveconstant C.

2. When adding a grain on site x, the expected number of topplings at sitey satisfies

limn

Vn(d) nVn(x, y, ) = G(x, y), (2.12)


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where G(x, y) is the Greens function of the simple random walk on theinfinite tree, i.e.

G(0, x) = C2|x|, (2.13)

and |x| is the generation number of x in the tree.

3. The correlations in the finite volume measures Vk can be estimated interms of the eigenvalues of a product of transfer matrices. This formalismis explained in detail in [DM (1990b)], section 5: Let f, g be two local func-tions whose dependence sets (see below a precise definition) are separatedby n generations. To estimate the truncated correlation function

Vk (f; g) =

fgdVk

f dVk

gdVk , (2.14)

consider the product of matrices

Mkn =

ni=1

1 +

k,ni 1 +

k,ni

1 2 + k,ni

, (2.15)

where k,ni [0, 1]. Let n,km (resp.

n,kM ) denote the smallest (resp.

largest) eigenvalue of Mkn . Then

Vk (f; g) C(f, g)n,km

n,kM. (2.16)

If f and g have dependence sets deep within Vk, then k,ni is very close

to one, and the correlations are governed by the maximal and minimal

eigenvalues of Mn =

2 21 3

n.

3 Main results

3.1 Notation, definitions

From now on, S denotes the infinite rootless binary tree, V S a finite subsetof S; V is the set of stable configurations in V, i.e. V = { : V {1, 2, 3}},and the set of all infinite volume stable configurations is = {1, 2, 3}S. Theset is endowed with the product topology, making it into a compact metricspace. For , |V is its restriction to V, and for , , VVc denotes theconfiguration whose restriction to V (resp. Vc) coincides with |V (resp. |Vc).As in the previous section, RV V is the set of all allowed (or recurrent)configurations in V, and we define

R = { : V S finite, |V RV}. (3.17)

A function f : IR is local if there is a finite V S such that |V = |Vimplies f() = f(). The minimal (in the sense of set ordering) such V is calleddependence set of f and is denoted by Df. A local function can be seen as afunction on V for all V Df and every function on V can be seen as a localfunction on . The set L of all local functions is uniformly dense in the set C()of all continuous functions on .

All along the paper, we use the following notion of limit by inclusion for afunction f on the finite subsets of the tree with values in a metric space (K, d):


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Definition 3.1 LetS = {V S, V finite}, and f : S (K, d). Then

limVS f(V) =

if for all > 0, there exists V0 S such that for all V V0, d(f(V), ) < .

Definition 3.2 A collection of probability measures V on V is a Cauchy netif for any local f and for any > 0 there exists V0 Df such that for anyV, V V0

|

f()V(d)

f()V(d)| .

A Cauchy net converges to a probability measure in the following sense:The mapping

: L IR; f (f) = limVS

f dV

defines a continuous linear functional on L (hence on C()) which satisfies(f) 0 for f 0 and (1) = 1. Thus by Riesz representation theoremthere exists a unique probability measure on such that (f) =

f d. We

denote V , and call this the infinite volume limit of V.We will also often consider an enumeration of the tree S, {x0, x1, . . . , xn, . . .},

and putTn = {x0, . . . , xn}. (3.18)

3.2 Thermodynamic limit of stationary measures

Theorem 3.1 The set R defined in (3.17) is an uncountable perfect set, i.e.

1. R is compact,

2. The interior ofR is empty,

3. For all R there exists a sequence n = , n R, converging to .

For , we denote by C3(0, ) the nearest neighbor connected cluster ofsites containing the origin and having height 3.

Theorem 3.2 The finite volume stationary measures V defined in (2.8) forma Cauchy net. Their infinite volume limit satisfies

1. (R) = 1,

2. is translation invariant and exponentially mixing,

3. ( : |C3(0, )| < ) = 1,

4.

|C3(0, )|(d) = .

Remark: Point 3. above remains true for the set C1(0, ), the nearest neighborconnected cluster of sites containing the origin and having height 1, and probablyalso for C2(0, ) but this we have not been able to prove.


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3.3 Infinite volume dynamics

The finite volume addition operators ax,V (cf. (2.5)) can be extended to via

ax,V : ; ax,V = (ax,V|V)VVc . (3.19)

Proposition 3.1

1. There exists a subset of R with () = 1 on which the limit

limVS

ax,V = ax (3.20)

exists, andax .

2. The measure of theorem 3.2 is invariant under the action ofax.

3. For every , ax(ay) = ay(ax), for all x, y S.

Part 2 implies that the infinite volume addition operators ax (cf. (3.20))define norm 1 operators on Lp(), for 1 p via

(axf)() = f(ax).

We now construct a Markov process on -typical infinite volume configura-tions which can be described intuitively as follows. Let : S (0, ); thisfunction will be the addition rate function. To each site x S we associatea Poisson proces Nt,x (for different sites these Poisson processes are mutuallyindependent) with rate (x). At the event times ofNt,x we add a grain at x,i.e. we apply the addition operator ax to the configuration. Then L

V introduced

in (2.7) generates a pure jump Markov process on . Indeed, this operator iswell-defined and bounded on any Lp() space by Proposition 3.1, which implies

Proposition 3.2 LV is the Lp() generator of the stationary Markov process

defined by

exp(tLV)f =

xV

aNt,xx f

dIP,

where IP denotes the joint distribution of the independent Poisson processesNt,x , and f L

p().

The following condition on the addition rate is crucial in our construction.Remember |x| is the generation number of x:

Summability Condition:xS

(x)2|x| < (3.21)

This condition ensures that the number of topplings at any site x Sremains finite during the addition process.

Theorem 3.3 If satisfies condition (3.21), then we have


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1. The semigroups SV(t) = exp(tLV) converge strongly in L

1() to a semi-group S(t).

2. S(t) is the L1() semigroup of a stationary Markov process {t : t 0}

on .

3. For anyf L,

limt0

S(t)f f

t= Lf =

xS

(x)[axf f],

where the limit is taken in L1().

Remarks.

1. In Proposition 4.1, we show that S(t) is a strongly continuous function of

.2. In Proposition 4.2, we show that condition (3.21) is in some sense optimal.

Theorem 3.4 The process {t : t 0} of Theorem 3.3admits a cadlag version(right-continuous with left limits).

The intuitive description of the process {t : t 0} is actually correctunder condition (3.21), i.e. the process has a representation in terms of Poissonprocesses:

Theorem 3.5 If satisfies condition (3.21), for IP almost every (, ) thelimit

limVS xV

aNt,x ()x = t

exists. The process {t : t 0} is a version of the process of Theorem 3.3, i.e.its L1() semigroup coincides with S(t).

Finally, we can slightly generalize Theorem 3.5 in order to define the dy-namics starting from a measure stochastically below . For , , if for all x S, (x) (x). A function f : IR is monotone if implies f() f(). Two probability measures and satisfy if for allmonotone functions,

f d

f d.

Theorem 3.6 Let . If satisfies condition (3.21), for IP almostevery (, ) the limit

limVS

xV

aNt,x ()x = t

exists. The process {t : t 0} is Markovian with 0 distributed according to.

Remark. The last Theorem implies that 1 can be taken as initial configu-ration.


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4 Proofs

This section is devoted to the proofs of the results described above. Some ofthem will be put in a slightly more general framework so that they can be ap-plied to other cases (where S is not a binary tree or where we have other additionoperators ax) as soon as the existence of a thermodynamic limit of the finitevolume stationary measures is guaranteed. The essential cause of difficulty isthe non-locality of the addition operators. The essential simplification is theabelian property which enables us to think of the ax as complex numbers ofmodulus one.

4.1 Thermodynamic limit of stationary measures

Proof of Theorem 3.1:

1,2. If R and , then R (by the burning algorithm |V |Vimplies that |V RV). Since 2 is in R (again by the burning algorithm), weconclude that R is uncountable. To see that R has empty interior, notice that if R, there does not exist x, y S nearest neighbors such that (x) = (y) = 1(that way, |{x,y} would be a forbidden subconfiguration). Finally R is closedas intersection of closed sets.

3. Let max be the maximal configuration, max(x) = 3 for all x S. If|V RV, then V(max)Vc R. Therefore any R containing an infi-nite number of sites x for which (x) = 3 has property 3 of Theorem 3.1. If R contains only a finite number of sites having height 1 or 2, then wechoose a sequence = {xn : n IN} {x S : (x) = 3 and (y) =3 for any neighbor of x} such that two elements of are never nearest neigh-bors, and |xn| is strictly increasing in n. We then define n(x) = (x) forx S\ {xk : 0 k n} and n(xk) = 2 for xk , k n + 1. These nbelong to R by the burning algorithm, and n .


We use Tn introduced in (3.18), but with the xi such that n m implies thatthe generation numbers satisfy |xn| |xm|. Then we have

|xn| log2 n. (4.22)

To prove that the probability measures V form a Cauchy net, it is sufficient toshow that for any local function f : IR we have

n

|

f dTn f dTn+1| < . (4.23)

We do it for f() = (x0) (a general local function can be treated in the sameway), by giving an upper bound of the difference

f dTn

f dTn+1 by a

truncated correlation function (cf. (2.14)). Then we estimate the latter by thetransfer matrix method (cf. Section 2.9, part 3). We abbreviate in what followsn = Tn , Rn = RTn .


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Lemma 4.1

|n+1[(x0)] n[(x0)]| Cn+1[(x0); I((xn+1) = 3)]

Proof: By the burning algorithm, every Rn can be extended to an elementof Rn+1 by putting (xn+1) = 3. Moreover

{|Tn : Rn+1, (xn+1) = 3} = Rn,

thus

n+1[(xn+1) = 3] =|Rn|

|Rn+1|, (4.24)

which yields

n((x0)) =

Rn1

|Rn|(x0)

=

Rn+1

1

|Rn+1|(x0)I((xn+1 = 3)

|Rn+1|

|Rn|

=

Rn+1

1

|Rn+1|(x0)I((xn+1) = 3)

1

n+1[(xn+1) = 3].

Therefore

|n+1((x0)) n((x0))| n+1[(x0); I((xn+1) = 3)]

n+1[(xn + 1) = 3]

The Lemma follows now from (4.24), and the fact that |Rn| grows like ecn forsome c log 2.

Recalling Section 2.9, part 3, we have

n[(x0); I((xn) = 3)] C|xn|,|xn|m

|xn|,|xn|M

(4.25)

Lemma 4.2

+n=1

|xn|,|xn|m

|xn|,|xn|M

< +

Proof: We abbreviate (n)m =

|xn|,|xn|m ,

(n)M =

|xn|,|xn|M , M(n) = M

n|xn|

, i =

|xn|,|xn|

i. Remember 0

|xn|,|xn|

i 1 is close to one for i n and n large. In

terms of the trace and the determinant of M(n) we have

(n)M =

1

2

Tr(M(n)) +

[Tr(M(n))]2 4det(M(n))

(n)m =

1

2

Tr(M(n))

[Tr(M(n))]2 4det(M(n))

.

Therefore,

limn

(n)m

(n)M

[Tr(M(n))]2

det(M(n))

= 1.


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To prove the Lemma we show that (cf. (4.22))

det(M(n))

[Tr(M(n))]2

( 4

9)|xn|. (4.26)

Use

det(M(n)) =

|xn|i=1

(1 + i)2,

and

Tr(M(n)) Tr

|xn|

i=1

1 + i 0

0 2 + i

=

|xn|i=1

(1 + i) +

|xn|i=1

(2 + i),

to estimate (for 1 i |xn|, 2(2 + i) 3(1 + i))

det(M(n))

[Tr(M(n))]2

1 + 2 |xn|

i=1

2 + i1 + i

+

|xn|i=1

2 + i1 + i

21

(1 + 2.(3/2)|xn| + (3/2)2|xn|)1

(4/9)|xn|.

For a general local function f, we have to replace |xn| by |xn| N0, whereN0 is the number of generations involved in the dependence set of f. Since f is

local, N0 is finite, hence the convergence in (4.23) is unaffected.

4.2 Infinite volume toppling operators

Definition 4.1 Given the finite volume addition operators ax,V (defined in(3.19)) acting on , we call a configuration normal if for every x Sthere exists a minimal finite set Vx() S such that for all V V Vx()

ax,V = ax,V.

In other words, for a normal , outside Vx(), no sites are affected by addinga grain at x. In our case, when a particle is added at some site x S, the

cluster of toppled sites coincides with the cluster C3(x, ) of sites having height3 including x, thus

Vx() = C3(x, ) eC3(x, ), (4.27)

where e denotes the exterior boundary. Notice that for a normal configuration, by definition,

ax() = limVS

ax,V() = ax,Vx()() (4.28)


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Proof of Proposition 3.1:

1. We show that there is a full measure set of normal configurations. From

(2.11) and Theorem 3.2,I(|C3(x, )| = n)d Cn

3/2.

Therefore concentrates on the set of configurations for which all the clustersC3(x, ) are finite, hence for which is normal. Moreover this set is closedunder the action of the addition operators ay, since (cf. (4.27))

C3(x, ay) Vx() Vy(). (4.29)

2. Choose > 0, pick a local function f, fix Vn S and n0 such that n n0implies

{ : Vx() Vn}

4f + 1 . (4.30)

This n0 exists since concentrates on normal configurations. We estimate

f(ax)d f()d

f(ax,Vn)d

f()d

+ 2f{ : Vx() Vn}

limm

f(ax,Vn)dVm

f()dVm+

2

2+ 2f lim

mVm (ax,Vn() = ax,Vm())

=

2+ 2f

1 lim

mVm(Vx() Vn)

= 2

+ 2f (1 (Vx() Vn)) .

In the last step we used that the indicator I(Vx() Vn) is a local function.3. Let , x, y S be two different sites and V Vx() Vx(ax)

Vx(ay). Since ax,V and ay,V commute, we have

ax(ay) = ax(ay,V) = ax,V(ay,V)

= ay,V(ax,V) = ay,V(ax) = ay(ax).

4.3 Infinite volume semigroupWe now turn to the proofs of Theorems 3.3 and 3.4.

Definition 4.2 We define the cluster of at x S as

C(x, ) = {y S : ay(x) = (x)}, (4.31)

and put

G(x, y) =

I(y C (x, ))d(). (4.32)


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Finally for : S [0, ), write

f =xS

(x)

(d)|f(ax) f()|,

B = {f : IR : f bounded, f < }.

Lemma 4.3 If xS

(x)G(y, x) < for all y S, (4.33)

then all local functions are in B.

Proof: Let f be a local function with dependence set Df. Then f(ax) = f()if for y Df, ax(y) = (y), i.e. x C(y, ):

f = xS

(x) |axf f|d=

xyDf C(y,)

(x)|axf f|d

2fxS

(x)

I(x yDf C(y, ))d

2fyDf

xS

G(y, x)(x) < .

The next lemma provides a link between G and the Greens function for

simple random walk on S, i.e., between conditions (4.33) and (3.21).Lemma 4.4

G(x, y) zx

G(y, z) = x,y + 3G(x, y),

where z x means that z and x are neighbors.

Proof: We have to estimate the probability that ax(y) = (y). If by addinga grain at x we influence y, this can only be achieved by the toppling of one ofthe nearest neighbor sites of y. Since concentrates on normal configurations,

(ax(y) = (y)) = limVS

(ax(y) = (y), Vx() Vy() V)

= limVS

limWS

W (ax,V(y) = (y), Vx() Vy() V)

= limVS

limWS

W (ax,W(y) = (y), Vx() Vy() V)

limWS

W (ax,W(y) = (y))

limWS

W ( z W, z y, nW(z , y , ) 1)

limWS

zy

dW()nW(z , y , )

=zy

G(z, y), (4.34)


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where we used (2.10),(2.12), (4.27) and (4.29).

The following Lemma finishes the proof of Theorem 3.3 and shows that B isa natural core for the domain of the generator of the infinite volume semigroup.

Lemma 4.5

1. For f B the net

SV(t)f = exp(tLV)f = exp

txV

(x)(ax I)

f (4.35)

converges in L1() (as V S) to a function S(t)f L1(). f S(t)f

defines a semigroup on B which is a contraction in both L1() and B

norms.

2. Under condition (4.33), the semigroup S

(t) corresponds to a unique Markovprocess on .

Proof: We denote by f the L1()-norm of f, and we abbreviate SV(t) =SV(t), S(t) = S

(t), LV = LV.

1. First note that SV(t) is well-defined on L1() by Proposition 3.2. By theabelian property (Proposition 3.1, part 3) we can write for V V S:

SV(t)f SV(t)f = (SV\V(t) I)SV(t)f

By Proposition 3.2, SV(t) is the semigroup of a stationary Markov process andhence a contraction on L1(). Therefore

SV(t)(SV\V(t) I)f (SV\V(t) I)f

= t0 LV

\VSV

\V(s)f ds

t0

LV\Vfds

t

xV\V

(x)

|(ax I)f|d 0 as V, V

S, (4.36)

where the last step follows from f B. Hence SV(t)f S(t)f in L1(). We

show that S(t)f B:xS

(x)

|S(t)f(ax) S(t)f()|(d)

xS

(x) S(t)|axf f|d=

xS

(x)|axf f|d = f.

Thus S(t) is also a contraction for the -norm. We finish with the semigroupproperty:

S(t)S(s)f = limVS

SV(t)[S(s)f]

= limVS

limWS

SV(t)SW(t)f,


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andS(t + s)f = lim

VS

SV(t)SV(s)f.

Then, since SV(t) is a contraction in L1(),

SV(t)SW(t)f SV(t)SV(s)f SW(s)f SV(s)f, (4.37)

By (4.36), the right hand side of (4.37) goes to zero as V, W S.2. If condition (4.33) is met, then B contains all local functions by Lemma4.3. Therefore, by contractivity the semigroup S(t) on B uniquely extendsto a semigroup of contractions on L1(). Since by Proposition 3.2, SV(t) is aMarkov semigroup, so is S(t), i.e. S(t)1 = 1, S(t)f 0 if f 0. Hence, byKolmogorovs theorem there is a unique Markov process with semigroup S(t).

Remark. When 1, condition (4.33) is equivalent to

xS

(d)I(x C (y, )) =

|C(y, )|(d) < +,

i.e., the clusters must be integrable under . For models which exhibit self-organized criticality, C(y, ) is usually a finite but critical percolation cluster,implying that

|C(y, )|d = (cf. Theorem 3.2 part 4, because C(y, )

eC3(y, )). Therefore this formalism breaks down for addition rate 1.The following Lemma proves Theorem 3.4.

Lemma 4.6 Under condition (4.33), the process {t : t 0} of Theorem 3.3is almost surely right-continuous, i.e.

IP limt0

d(t, 0) = 0, (4.38)where IP is its path-space measure, and the distance d is defined below (in(4.41)).

Proof: Pick a function : S (0, 1) such thatxS

(x) = 1, (4.39)

and x,yS

(x)G(x, y)(y) < (4.40)

The distance

d(, ) =xS |(x) (x)|(x) (4.41)

generates the product topology. Denote by IE the expectation w.r.t. IP. Forfy() = (y),

fy(t) fy(0) =

t0

Lfy(s)ds + Myt ,

where Myt is a centered martingale with quadratic variation

IE

(Myt )2

= IE

t0

(Lf2y (s) 2fy(s)Lfy(s))ds

. (4.42)


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Using stationarity of s and

d|Lg| 2g

xS

yDg

(x)G(y, x),

for a local bounded function on , we obtain from (4.42)

IE

(Myt )2

CtxS

(x)G(y, x).

Now we can estimate

IP

s t :

yS

|s(y) 0(y)|(y)

IPt

0dsyS

|Lfy(s)|(y) /2

+ IP

sup0st

yS Mys (y) /2

(12t/)x,yS

(x)G(y, x)(y) + (2/)2IE

yS

Myt (y)

2

(12t/)x,yS

(x)G(y, x)(y) + (2/)2IE

yS

(Myt )2(y)

tCx,yS

(x)G(y, x)(y).

Here we used Markovs and Doobs inequalities in the second step and the

Cauchy-Schwarz inequality combined with (4.39) in the third step. The result(4.38) follows.

4.4 Poisson representation

In this section we prove Theorems 3.5 and 3.6. Intuitively it is clear from theabelian property that the process of which we showed existence in the previ-

ous subsection can be represented as

xS aNt,xx , where Nt,x are independent

Poisson processes of intensity (x).We take Tn as in (3.18). We say that the product

xS a

nxx exists if for

every y S there exists Ny such that for all m, n Ny xTn

anxx

(y)

xTm

anxx

(y)

= 0.This is equivalent to the convergence of the sequence

xTn

anxx in the producttopology.

Lemma 4.7 Under condition (4.33), the productxS

aNt,xx = t


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exists for -almost every realization of Nt,x and almost every . The process{t : t 0} is a version of the Markov process of Lemma 4.5.

Proof: Choose a realization of Nt,x such thatxS

Nt,x G(x, y) < (4.43)

for every y. This happens with probability one by condition (4.33). Define for

Tn(t) =xTn

aNt,xx . (4.44)

Under , Tn(t) is stationary in n and t. We have

(Tn(t))(y) (Tn+1(t))(y) 1

aNt,xn+1

xn+1 Tn(t)

(y) (Tn(t))(y)

(d)=

aNt,xn+1

xn+1

(y) (y)

(d)

Nt,xn+1j=1

ajxn+1 (y) aj1xn+1 (y)(d)

6Nt,xn+1 G(xn+1, y).

In the second and last steps we used the invariance of under ax. By theBorel Cantelli Lemma, (4.43) implies that for almost every realization of Nt,x

n0 : n n0 (Tn(t))(y) = (Tn0 (t))(y)

= 1.

This proves -a.s. convergence of the product. To see that t is a version of theMarkov process with semigroup S(t), combine Proposition 3.2 with Theorem3.3, part 1 to get, for any local function f,

d

dIPf(t) S(t)f

= 0.In the preceding argument we used a particular enumeration of the countableset S. But changing it gives again a process with semigroup S(t). Thereforethe limiting process will not depend (up to sets of measure zero) on the chosenenumeration of S.


For and y S we have the relation (remember (4.44))

V(t)(y) = (y) + I(y V)xV

Nt,x ntV(y), (4.45)

where ntV(x), an integer valued random variable, is the number of topplings atsite x in the time interval [0, t], when sand is added in V. For Tn defined in (3.18)we will first prove that ntTn increases IP almost surely to an L

1(IP) random


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variable nt, interpreted as the number of topplings in [0, t] when we add grainsaccording to Nt,x . By the abelian property the sequence n

tTk

(0) is increasing in

k. The following estimate is similar to (4.34)

( IP) ( |ntTk(0) ntTk+1(0)|

= ( IP)

ntxk+1(0)

1

ntxk+1(0)(d) IP(d)

1

limVS

ntxk+1(0)I(Vxk+1() V0() V)(d) IP(d)

1

limWS

ntW(xk+1, 0, )W(d) IP(d)

1

t(xk+1)G(0, xk+1). (4.46)

In the fourth line, ntW(xk+1, 0, ) denotes the number of topplings up to time tat site 0 W by adding grains at site xk+1 W. By the Borel Cantelli Lemma,condition (3.21) implies the a.s. convergence of ntTk (0), and analogously of everyntTk(x). Pick (, ) such that n

tTk

(, ) converges, i.e. such that supk ntTk

(, ) =nt(, ) is finite (indeed, ntTk(, ) is an integer). If

, then ntTk(, )

ntTk(, ) because we can obtain from by adding sand at sites x Sfor which

(x) < (x) thereby increasing the number of topplings. We thus conclude thatthe convergence ofntTk(, ) implies the convergence ofn

tTk

(, ) for all .Now let in the FKG sense. There is a coupling IP12 of IP and

IP such that

IP12 (((1, 1), (2, 2)) : 1 = 2, 1 2) = 1,

i.e. we use the same Poisson events and couple

and according to the optimalcoupling (see [Strassen (1965)]). Then

( IP)

ntTk (, ) nt(, )

= IP12

ntTk (1, 1) n

t(1, 1)

= IP12

ntTk(1, 1) nt(1, 1), n

tTk(2, 2) n

t(2, 2), 1 = 2, 1 2

IP12

ntTk(2, 2) nt(2, 2)

= ( IP)

ntTk(, ) n

t(, )

= 1.

This shows the IP-almost sure convergence ofntTk , hence by (4.45) the prod-

uct

xS aNt,x ()x converges IP almost surely.

As a further result we show that the semigroup S(t) is continuous as a

function of the addition rate . We define

1 = { : S [0, ) : =xS

(x)G(0, x) < }.

It is a complete metric space (as a closed subset of a Banach space) with theproperty: If n 1, n (pointwise), and 1 then n in 1.

Proposition 4.1 The semigroup S(t) of Theorem 3.3is a strongly continuous function of, i.e. ifn in 1, then for any local function f, Sn(t)f S(t)f.


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Proof: Let nt = limk ntTk

be the number of topplings in [0, t] from sandaddition at rate . In the proof of Theorem 3.5 we have shown that this random

variable is IP almost surely well defined and, after taking limits in (4.45),satisfies

t = 0 + Nt n

t, (4.47)

where Nt = limVS

xV Nt,x . Note that if 1 2, the Poisson processes

Nt1 and Nt2

can be coupled in such a way that for all x S, Nt,x1 Nt,x2

, and

hence, by the abelian property, nt1(x) nt2

(x). Consider a coupling of the

four Poisson processes Nt, Ntn, N

tn and N

tn under which the inequalities

X1(t) X2(t), X2(t) X3(t), X3(t) X4(t), are satisfied with probability one.Let IP denote the law of the marginal (X1, X4). We have, by a reasoning similarto (4.46),

d

IE|n

t

n(0) n

t

(0)|

d

IE

n

t

n(0) n

t

n(0)

+ IE

ntn(0) ntn(0)

+ IE

ntn(0) n

t(0)

txS

|n(x) (x)|G(0, x), (4.48)

which tends to zero for n in 1. Take now a local function f, and denoteDf = Df eDf,

|Sn(t)(f) S(t)(f)| IP

ntn(x) = nt(x) for some x Df

xDf

IE|nt

(x) nt

n(x)|.

Combining this with (4.48) concludes the proof.

One might ask whether we can go beyond condition (3.21), which essentiallyguarantees that the expected number of topplings stays finite in the additionprocess. In the following proposition we show that it is impossible to keepintegrable toppling numbers and rate 1 addition. The relation (4.49) shouldbe regarded as the infinitesimal version of (4.47), where (x) replaces the rate(x). We then show that has to depend on x.

Proposition 4.2 Let : S {0, 1} be a stationary and ergodic random fielddistributed according to . Denote by (0)(d) = its density. Supposethere exists a measurable transformation T : {0, 1}S which satisfies theconditions

1. The measure of Theorem 3.2 is invariant under T(, ) for any .

2.T(, )(x) = (x) + (x) n(,,x), (4.49)

with n(,,.) L1() for almost every .

Then, = 0.


23/24


Proof: Taking expectation over in (4.49) gives

(, x) = (x), (4.50)

where (, x) =

n(,,x)(d). By stationarity of and , (, x) is astationary random field. Let (xt : t 0) denote continuous time simple randomwalk on S, starting at 0. From (4.50),

IE(, xt) = (, 0) + IE

t0

(xs)ds.

Divide this last line by t and let t +. As is ergodic (making the last termequal to ) and as the process (, xt) is stationary, we conclude that = 0.

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[Chung (1960)] Chung, K.L., Markov Chains with Stationary Transition Prob-abilities, Springer-Verlag, Berlin-Gottingen, Heidelberg, 1960.

[DR (1989)] Dhar, D. and Ramaswamy, R., Exactly Solved Model of Self-Organized Critical Phenomena, Phys. Rev. Lett. 63, 16591662 (1989).

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

[DM (1990b)] Dhar, D. and Majumdar, S.N., Abelian Sandpile Models on theBethe Lattice, J. Phys. A 23, 43334350 (1990).

[Dhar (1999)] Dhar, D., The Abelian Sandpiles and Related Models, Physica A263, 425 (1999).

[IP (1998)] E.V. Ivashkevich, Priezzhev, V.B., Introduction to the sandpilemodel, Physica A 254, 97116 (1998).

[Liggett (1980)] Liggett, T.M., The Long Range Exclusion Process, Ann. Prob.8, 861889 (1980).

[Liggett (1985)] Liggett, T.M., Interacting Particle Systems, Springer, Berlin,Heidelberg, New York, 1985.

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

[Chen (1992)] Mu Fa Chen, From Markov Chains to Non-Equilibrium ParticleSystems, World Scientific, 1992.

[Priezzhev (1994)] Priezzhev, V.B., Structure of Two Dimensional Sandpile.I.Height Probabilities, J. Stat. Phys. 74, 955979 (1994).


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

[Strassen (1965)] Strassen, V., The existence of probability measures with givenmarginals, Ann. Math. Statist. 36, 423439 (1965).

[Toom (1990)] Toom, A.L., Vasilyev, N.B., Stavskaya, O.N., Mityushin, L.G.,Kurdyumov, G.L., Pirogov, S.A., Discrete local Markov Systems. In: Do-brushin, R.L., Kryukov V.I., Toom, A.L. (eds.), Stochastic cellular systems:ergodicity, memory, morphogenesis, Manchester University Press, 1182,1990.

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

Adresses:

C.M.: Instituut voor Theoretische Fysica, K.U.Leuven, Celestijnenlaan 200D,B-3001 Leuven, Belgium - email: [email protected].: On leave from Instituut voor Theoretische Fysica, K.U. Leuven, Celestij-nenlaan 200D, B-3001 Leuven, Belgium - email: [email protected].: CNRS, UMR 6085, Universite de Rouen, 76821 Mont-Saint-Aignan cedex,France. - email: [email protected]

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