THE ABELIAN SANDPILE MODEL ON AN INFINITE TREE KU Leuven, TU Eindhoven and CNRS ...

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The Abelian sandpile model on an infinite tree
Citation for published version (APA):
Maes, C., Redig, F. H. J., & Saada, E. (2002). The Abelian sandpile model on an infinite tree. The Annals of
Probability, 30(4), 2081-2107. https://doi.org/10.1214/aop/1039548382
DOI:
10.1214/aop/1039548382
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Published: 01/01/2002
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TheAnnalsofProbability
2002,Vol.30,No.4,2081–2107
THE ABELIAN SANDPILE MODEL ON AN INFINITE TREE
BY CHRISTIAN MAES, FRANK REDIG AND ELLEN SAADA
K.U.Leuven,T.U.EindhovenandC.N.R.S.,Rouen
WeconsiderthestandardAbeliansandpileprocessontheBethelattice.
We show the existence of the thermodynamic limit for the finite volume
stationary measures and the existence of a unique infinite volume Markov
processexhibitingfeaturesofself-organizedcriticality.
1. Introduction. Markovprocessesforspatiallyextendedsystemshavebeen
around for about 30 years now and interacting particle systems have become
a branch of probability theory with an increasing number of connections with
the natural and human sciences. While standard techniques and general results
have been collected in a number of books such as Liggett (1985), Chen (1992)
and Toom (1990) and are capable of treating the infinite volume construction
for stochastic systems with locally interacting components, some of the most
elementary questions for long range and nonlocal dynamics have remained wide
open. We have in mind the class of stochastic interacting systems that during the
last decade have invaded the soft condensed matter literature and are sometimes
placedunderthecommondenominatorofself-organizingsystems.
SincetheappearanceofBak,TangandWiesenfeld(1988),theconceptofself-
organized criticality (SOC) has excited much interest, and has been applied in
a great variety of domains [see, e.g., Turcotte (1999) for an overview]. From
the mathematical point of view, the situation is, however, quite unsatisfactory.
The models exhibiting SOC are in general very boundary condition dependent
[especially the Bak–Tang–Wiesenfeld (BTW) 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 the point of view of interacting
particle systems no standard theorems are at our 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 the one-dimensional BTW model [see
Maes,Redig,SaadaandVanMoffaert(2000)]orthelong-rangeexclusionprocess
[seeLiggett(1980)].Ontheotherhand,tomakemathematicallyexactstatements
about SOC, it is necessary to have some kind of infinite volume limit, both for
staticsandfordynamics.
InthispaperwecontinueourstudyoftheBTWmodelforthecaseoftheBethe
lattice;thisistheAbeliansandpilemodelonaninfinitetree.Forthissystem,many
ReceivedDecember2000;revisedFebruary2002.
AMS2000subjectclassifications.Primary82C22;secondary60K35.
Keywordsandphrases.Sandpile dynamics, nonlocal interactions, interacting particle systems,
thermodynamiclimit.
2081
2082 C.MAES,F.REDIGANDE.SAADA
exact results were obtained in [Dhar and Majumdar (1990)]. In contrast to the
one-dimensional case this system has a nontrivial stationary measure. We show
here that the finite volume stationary measures converge to a unique measure µ
which is not Dirac and exhibits all the properties of a SOC state. We then turn
to the construction of a stationary Markov process starting from this measure µ.
The main difficulty to overcomeis the strong nonlocality:addinga grain atsome
lattice site x caninfluencetheconfigurationfarfrom x.In facttheclusterofsites
influencedbyaddingatsomefixedsitehastobethoughtofasacriticalpercolation
cluster which is almost surely finite but not of integrable size. The process we
constructis intuitively describedas follows:ateachsite x of the Bethelattice we
have an exponential clock which rings at rate ϕ(x). At the ringing of the clock
weaddagrainatx.Dependingontheadditionrate ϕ(x),weshowexistenceofa
stationary Markov processwhich correspondsto this description. We also extend
(cid:1)
thisstationarydynamicstoinitialconfigurationswhicharetypicalforameasureµ
thatisstochasticallybelowµ.
The paper is organized as follows. In Section 2 we introduce standard results
onfinitevolumeAbeliansandpilemodelsandsummarizesomespecificresultsof
[Dhar and Majumdar (1990)] for the Bethe lattice which we need for the infinite
volume construction. In Section 3 we present the results on the thermodynamic
limit of the finite volume stationary measures and on the existence of infinite
volume Markovian dynamics. Section 4 is devoted to proofs and contains some
additionalremarks.
2. Finite volume Abelian sandpiles. In this section we collect some results
on Abelian sandpiles on finite graphs which we will need later on. Most of these
results are contained in the review paper by Dhar (1999) or in Ivashkevich and
Priezzhev(1998).
2.1. Toppling matrix. Let V denote a finite set of sites. We will always
supposethatV is anearestneighborconnectedsubsetofZd orofT ,theinfinite
d
homogeneoustreeofdegreed+1.StartingfromSection3,wespecifytothetree.
A matrix (cid:8)=((cid:8)x,y)x,y∈V indexed by the elements of V is called a toppling
matrixifthefollowinghold:
1. forallx,y∈V,x(cid:5)=y,(cid:8) =(cid:8) ≤0;
x,y y,x
2. forallx∈V,(cid:8) ≥1;
(cid:1)x,x
3. f(cid:1)orallx∈V, y∈V (cid:8)x,y ≥0;
4. x,y∈V (cid:8)x,y >0.
The fourth condition ensures that there are sites (so-called dissipative sites) for
whichtheinequalityinthethirdconditionisstrict.Thisisfundamentalforhaving
awell-definedtopplingrulelateron.
ABELIANSANDPILEMODEL 2083
StartingfromSection3ofthispaperwewillchoose(cid:8)tobethelatticeLaplacian
withopenboundaryconditions.Moreexplicitly,
(cid:2)
2d, ifV ⊂Zd,
(cid:8) =
x,x d+1, ifV ⊂T ,
d
(1)
(cid:8) =−1 ifx andy arenearestneighbors.
x,y
ThedissipativesitesthencorrespondtotheboundarysitesofV.Theresultsonthe
finite volume Abelian sandpile in this section remain valid for a generaltoppling
matrix(cid:8).
2.2. Configurations. A height configuration η is a mapping from V to N=
{1,2,...} assigning to each site a natural number η(x)≥1 (“the number of sand
grains” at site x). A configuration η ∈ NV is called stable if, for all x ∈ V,
η(x)≤(cid:8) .Otherwiseηisunstable.Wedenoteby(cid:14) thesetofallstableheight
x,x V
configurations.Forη∈NV andV(cid:1)⊂V,ηV(cid:1) denotestherestrictionofη toV(cid:1).
2.3. Toppling rule. The toppling of a site x corresponding to the toppling
matrix(cid:8)isthemapping
T :NV ×V →NV
x
definedby
(cid:2)
η(y)−(cid:8) , ifη(x)>(cid:8) ,
(2) T (η)(y)= x,y x,x
x
η(y), otherwise.
In words, site x topples if and only if its height is strictly larger than (cid:8) , by
x,x
transferring −(cid:8) grains to site y (cid:5)=x and losing (cid:8) grains. Toppling rules
x,y x,x
commute on unstable configurations. This means, for x,z ∈ V and η such that
η(x)>(cid:8) andη(z)>(cid:8) ,
x,x z,z
(3) T T (η)=T T (η).
x z z x
For η ∈NV, we say that ζ ∈(cid:14) arises from η by toppling if there exists an
V
n-tuple(x ,...,x )ofsitesinV suchthat
1 n
(cid:3) (cid:5)
(cid:4)n
(4) ζ = T (η).
xi
i=1
Thetopplingtransformationisthemapping
T :NV →(cid:14)
V
defined by the requirement that T (η) arises from η by toppling. The fact that
stabilization of an unstable configuration is always possible follows from the
existence of dissipative sites (only a finite number of sites have to be toppled a
2084 C.MAES,F.REDIGANDE.SAADA
finitenumberoftimes).ThusonlythefactthatT iswelldefinedislesstrivial:one
hastoproveherethatforagivenunstableconfigurationeverypossiblestabilization
makesthesamesitestopplethesamenumberoftimes.Moreover,by(3)theorder
of the T in the product (4) is not important. A complete proof can be found in
xi
Meester(2002).
2.4. Addition operators. For η∈NV and x ∈V, let ηx denote the configura-
tion obtained from η by adding one grain to site x, that is, ηx(y)=η(y)+δ .
x,y
Theadditionoperatordefinedby
(5) a :(cid:14) →(cid:14) , η(cid:14)→a η=T (ηx)
x V V x
represents the effect of adding a grain to the stable configuration η and letting a
stableconfigurationarisebytoppling.BecauseT iswelldefined,thecomposition
ofadditionoperatorsiscommutative:forallη∈(cid:14) , x,y∈V,
V
a (a η)=a (a η).
x y y x
2.5. Finite volume dynamics. Let p denote a nondegenerate probability
(cid:1)
measure on V, that is, numbers px, 0<px <1, with x∈V px =1. We define a
discretetimeMarkovchain{η :n≥0}on(cid:14) bypickingapointx ∈V according
n V
to p at each discrete time step and applying the addition operator a to the
x
configuration.ThisMarkovchainhasthetransitionoperator
(cid:6)
(6) Pf(η)= p f(a η).
x x
x∈V
We can equally define a continuous time Markov process {η :t ≥ 0} with
t
infinitesimalgenerator
(cid:6)
(7) Lϕf(η)= ϕ(x)[f(a η)−f(η)],
x
x∈V
generatingapurejumpprocesson(cid:14) ,withadditionrateϕ(x)>0atsitex.
V
2.6. Recurrent configurations, stationary measure. We see here that the
Markov chain {η ,n≥0} has only one recurrent class and its stationary measure
n
istheuniformmeasureonthatclass.
LetuscallR thesetofrecurrentconfigurationsfor{η :n≥0},thatis,those
V n
for which P (η =η infinitelyoften)=1, where P denotes the distribution of
η n η
{η :n≥0} starting from η =η∈(cid:14) . In the following propositionwe list some
n 0 V
properties of R . For the sake of completeness we include the proof which can
V
alsobefoundinMeester(2002).
PROPOSITION 2.1. (i) RV containsonlyonerecurrentclass.
(ii) The addition operators a generate an Abelian group G of permutations
x
ofR .
V
ABELIANSANDPILEMODEL 2085
(iii) ThegroupGactstransitivelyonR .Inparticular|G|=|R |.
V V
(iv) |R |=det(cid:8).
V
PROOF. (i) Wewrite η(cid:29)→ζ ifintheMarkovchainζ canbereachedfromη
withpositiveprobability.Sincesandisaddedwithpositiveprobabilityonallsites
(p >0),themaximalconfigurationη definedby
x max
η (x)=(cid:8)
max x,x
can be reached from any other configuration. Hence, if η∈R , then η(cid:29)→η ;
V max
thereforeη ∈R andη (cid:29)→η [see,e.g.,Chung(1960),page19].
max V max
(ii) Fixη∈R ;thenthereexistn ≥1suchthat
V y
(cid:4)
anyη=η
y
y∈V
and
(cid:4)
g =anx−1 any
x x y
y∈V,y(cid:5)=x
satisfies(a g )(η)=(g a )(η)=η.Theset
x x x x
Rx ={ζ ∈R :(a g )(ζ)=ζ}
V x x
isclosedundertheactionofa ,containsη,hencealsoη :itisarecurrentclass.
x max
By (i), Rx =R , a g is the neutral element e and g =a−1 if we restrict a
V x x x x x
toR .
V
(iii) Fix ζ ∈ R and put :G → R ;g (cid:14)→ g(ζ). As before (G) is
V ζ V ζ
a recurrent class; hence (G) = R . If for g,h ∈ G, (g) = (h), then
ζ V ζ ζ
gh−1(ζ)=ζ, and by commutativity gh−1(g(cid:1)ζ)=g(cid:1)ζ for any g(cid:1) ∈G. Therefore
gh−1(ξ)=ξ for all ξ ∈R ; thus g =h. This proves that is a bijection from
V ζ
GtoR .
V
(iv) Adding (cid:8) particles at a site x ∈V makes the site topple, and −(cid:8)
x,x x,y
particlesaretransferredtoy.Thisgives
(cid:4)
a(cid:8)x,x = a−(cid:8)x,y.
x y
y(cid:5)=x
OnR thea canbeinvertedandweobtaintheclosurerelation
V x
(cid:4)
(8) a(cid:8)x,y =e.
y
y∈V
Write
(cid:7) (cid:8)
(cid:8)ZV = (cid:8)n:n=(ny)y∈V ∈ZV ,
where
(cid:6)
((cid:8)n) = (cid:8) n .
x x,y y
y∈V
2086 C.MAES,F.REDIGANDE.SAADA
Consider
(cid:4)
(9) :ZV →G:n(cid:14)→ anx.
x
x∈V
Themap isahomomorphismfromZV ontoG,thatis, (n+m)= (n) (m)
and (ZV)=G.Therefore,G isisomorphicto thequotientZV/Ker( ). By(8),
(cid:8)ZV is contained in Ker( ). Conversely, let n∈Ker( ), and put n=n+−n−,
where n+ = max{n ,0}, n− = max{−n ,0}. Since (n) = e, adding to a
x x x x
recurrentη∈R accordingto n+ hasthesameeffectasaddingaccordingto n−.
V
Therefore,thereexistk+,k−∈(Z+)V,ζ ∈R suchthat
V
η+n+−(cid:8)k+=ζ =η+n−−(cid:8)k−
andweconcluden∈(cid:8)ZV.ThisshowsthatGisisomorphicto ZV/(cid:8)ZV andthe
lattergrouphascardinalitydet(cid:8). (cid:1)
AsaconsequenceofthegrouppropertyofG,theuniquestationarymeasureis
uniformonR .
V
COROLLARY 2.1. (i) Themeasure
(cid:6)
1
(10) µ = δ
V |R | η
η∈RV V
is invariant under the action of a , x ∈ V (δ is the Dirac measure on
x η
configurationη).
(ii) OnL2(µ )theadjointofa is
V x
(11) a∗=a−1.
x x
REMARK. This shows that µV is invariant under the Markov processes
generatedby(6)and(7).
2.7. Allowed configurations. Given a configuration η ∈(cid:14) , we say that its
V
restrictionη toanonemptysubsetW ⊂V isaforbiddensubconfigurationif,for
W
allx∈W,
(cid:6)
η(x)≤ (−(cid:8) ).
x,y
y∈W,y(cid:5)=x
A configuration η ∈ (cid:14) is called allowed if it does not contain a forbidden
V
subconfiguration.WedenotebyA thesetofallallowedconfigurations.
V
PROPOSITION 2.2.
A =R .
V V
ABELIANSANDPILEMODEL 2087
It is easy to see that toppling or adding cannot create a forbidden subconfigu-
ration, which immediately implies A ⊃R . For a proof that A =R using
V V V V
spanningtrees seeIvashkevichandPriezzhev(1998);a directproofcanbe found
in Meester (2002). For a generalization to nonsymmetric toppling matrices, see
Speer(1993).
The property of having a forbidden subconfiguration in W ⊂V only depends
on the heights at sites x ∈ W. Therefore η ∈ R implies η ∈ R . This
V W W
“consistency”propertyenablesustodefineallowedconfigurationsoninfinitesets.
2.8. Expected toppling numbers. For x,y ∈ V and η ∈ (cid:14) , let n (x,y,η)
V V
denote the number of topplings at site y ∈V by adding a grain at x ∈V, that is,
thenumberoftimeswehavetoapplytheoperatorT tostabilizeηx.Define
y
(cid:9)
(12) G (x,y)= µ (dη)n (x,y,η).
V V V
Writing downbalancebetweeninflowandoutflowatsitey,oneobtains[cf.Dhar
(1990)]
(cid:6)
(cid:8) G (z,y)=δ ,
x,z V x,y
z∈V
whichyields
G (x,y)=((cid:8))−1.
V x,y
InthelimitV ↑S (whereS isZd ortheinfinitetree),G convergestotheGreen’s
V
functionofthesimplerandomwalkonS.
2.9. Some specific resultsfor the tree. When V is a binary tree of n genera-
n
tions,manyexplicitresultshavebeenobtainedinDharandMajumdar(1990).We
summarizeheretheresultsweneedfortheconstructionininfinitevolume.
1. When adding a grain on a particular site 0∈V of height 3, the set of toppled
n
sitesistheconnectedclusterC (0,η)ofsitesincluding0havingheight3.This
3
clusteris distributed asa randomanimal(i.e., its distribution only dependson
itscardinality,notonitsform).Moreover,
(cid:10) (cid:11)
(13) lim µ |C (0,η)|=k (cid:23)Ck−3/2
n↑∞ Vn 3
as k goes to infinity. The notation (cid:23) means that if we multiply the left-hand
sideof(13)byk3/2,thenthelimitk→∞issomestrictlypositiveconstantC.
2. When adding a grain on site x, the expected number of topplings at site y
satisfies
(cid:9)
(14) lim µ (dη)n (x,y,η)=G(x,y),
n↑∞ Vn Vn
2088 C.MAES,F.REDIGANDE.SAADA
whereG(x,y)istheGreen’sfunctionofthesimplerandomwalkontheinfinite
tree,thatis,
(15) G(0,x)=C2−|x|,
and|x|isthe“generationnumber”ofx inthetree.
3. Thecorrelationsinthefinitevolumemeasuresµ canbeestimatedintermsof
Vk
the eigenvaluesof a productof transfermatrices. This formalism is explained
in detail in Dhar and Majumdar (1990), Section 5. Let f,g be two local
functionswhosedependencesets(seebelowaprecisedefinition)areseparated
byngenerations.Toestimatethetruncatedcorrelationfunction
(cid:9) (cid:9) (cid:9)
(16) µ (f;g)= fgdµ − f dµ gdµ ,
Vk Vk Vk Vk
considertheproductofmatrices
(cid:3) (cid:5)
(cid:4)n 1+γk,n 1+γk,n
(17) Mk = i i ,
n 1 2+γk,n
i=1 i
where γk,n ∈[0,1]. The meaning of γk,n is explainedin Dhar and Majumdar
i i
(1990), but we will only use the fact 0≤γk,n ≤1 in Lemma 4.1 below. Let
i
λn,k (resp.λn,k)denotethesmallest(resp.largest)eigenvalueofMk.Then
m M n
λn,k
(18) µ (f;g)≤C(f,g) m .
Vk n,k
λ
M
For sites i far from the boundary of V , that is, for fixed i and n, in the limit
k
k→∞,γk,ntendsto1,andthecorrelationsbetweenlocalfunctionsinthelimit
i
V →S are then governed by the maximal and minimal eigenvaluesof M =
(cid:12)k (cid:13) n
2 2 n
. We shall need the estimate of a local function with a function living
1 3
ontheboundaryofV ;thereforewehavetousethefullexpression(17),(18).
k
3. Mainresults.
3.1. Notation,definitions. Fromnowon,S denotestheinfiniterootlessbinary
tree,V ⊂S afinitesubsetofS;(cid:14) isthesetofstableconfigurationsinV,thatis,
V
(cid:14) ={η:V →{1,2,3}}, and the set of all infinite volume stable configurations
V
is (cid:14)={1,2,3}S.Theset(cid:14) isendowedwith theproducttopology,makingitinto
a compact metric space. For η ∈(cid:14), η is its restriction to V, and for η,ζ ∈(cid:14),
V
ηVζVc denotesthe configurationwhoserestriction to V (resp. Vc) coincideswith
ηV (resp. ζVc). As in the previous section, RV ⊂(cid:14)V is the set of all allowed (or
recurrent)configurationsinV,andwedefine
(19) R={η∈(cid:14):∀V ⊂S finite, η ∈R }.
V V
ABELIANSANDPILEMODEL 2089
Afunctionf :(cid:14)→RislocalifthereisafiniteV ⊂S suchthatη =ζ implies
V V
f(η) = f(ζ). The minimal (in the sense of set ordering) such V is called the
dependence set of f and is denoted by D . A local function can be seen as a
f
functionon (cid:14) forall V ⊃D , andeveryfunction on (cid:14) canbeseenasalocal
V f V
functionon(cid:14).ThesetLofalllocalfunctionsisuniformlydenseinthesetC((cid:14))
ofallcontinuousfunctionson(cid:14).
Througout the paper, we use the following notion of limit by inclusion for a
functionf onthefinitesubsetsofthetreewithvaluesinametricspace(K,d):
DEFINITION 3.1. LetS={V ⊂S, V finite}andf :S→(K,d).Then
limf(V)=κ
V↑S
if,forallε>0,thereexistsV ∈S suchthat,forallV ⊃V ,d(f(V),κ)<ε.
0 0
DEFINITION 3.2. AcollectionofprobabilitymeasuresνV on(cid:14)V isaCauchy
net if, for any local f and for any ε>0, there exists V ⊃D such that, for any
0 f
V,V(cid:1)⊃V ,
0
(cid:14)(cid:9) (cid:9) (cid:14)
(cid:14) (cid:14)
(cid:14)(cid:14) f(η)νV(dη)− f(η)νV(cid:1)(dη)(cid:14)(cid:14)≤ε.
A Cauchynetconvergesto a probability measureν in the following sense:the
mapping
(cid:9)
:L→R, f (cid:14)→ (f)= lim f dν
V
V↑S
defines a continuous linear functional on L [hence on C((cid:14))] which satisfies
(f) ≥ 0 for f ≥ 0 and (1) = 1. Thus by the Riesz representation theorem
(cid:15)
thereexistsauniqueprobabilitymeasureon(cid:14)suchthat (f)= f dν.Wewrite
ν →ν andcallthisν theinfinitevolumelimit ofν .
V V
We will also often consider an enumeration of the tree S, {x ,x ,...,x ,...},
0 1 n
andput
(20) T ={x ,...,x }.
n 0 n
3.2. Thermodynamiclimitofstationarymeasures.
THEOREM3.1. ThesetRdefinedin(19)isaperfectset;thatis,thefollowing
hold:
(i) R iscompact.
(ii) TheinteriorofR isempty.
(iii) Forallη∈R thereexistsasequenceη (cid:5)=η,η ∈R,convergingtoη.
n n