University of Groningen Abelian Sandpiles and the Harmonic Model Schmidt, Klaus

40 Pages·2009·0.65 MB·English

University of Groningen

Abelian Sandpiles and the Harmonic Model
Schmidt, Klaus; Verbitskiy, Evgeny
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Communications in Mathematical Physics
DOI:
10.1007/s00220-009-0884-3
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Commun.Math.Phys.292,721–759(2009) Communicationsin
DigitalObjectIdentifier(DOI)10.1007/s00220-009-0884-3 Mathematical
Physics
Abelian Sandpiles and the Harmonic Model
KlausSchmidt1,2,EvgenyVerbitskiy3,4
1 MathematicsInstitute,UniversityofVienna,Nordbergstrasse15,A-1090Vienna,Austria.
E-mail:[email protected]
2 ErwinSchrödingerInstituteforMathematicalPhysics,Boltzmanngasse9,A-1090Vienna,Austria
3 PhilipsResearch,HighTechCampus36(M/S2),5656AE,Eindhoven,TheNetherlands.
E-mail:[email protected]
4 DepartmentofMathematics,UniversityofGroningen,POBox407,9700AK,Groningen,TheNetherlands
Received:15January2009/Accepted:14April2009
Publishedonline:15August2009–©TheAuthor(s)2009.Thisarticleispublishedwithopenaccessat
Springerlink.com
Abstract: We present a construction of an entropy-preserving equivariant surjective
mapfromthed-dimensionalcriticalsandpilemodeltoacertainclosed,shift-invariant
subgroupofTZd (the‘harmonicmodel’).Asimilarmapisconstructedforthedissipative
abelian sandpile model andisusedtoprove uniqueness and theBernoulliproperty of
themeasureofmaximalentropyforthatmodel.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
1.1 Fourmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
1.2 Outlineofthepaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
2. APotentialFunctionandits(cid:1)1-Multipliers . . . . . . . . . . . . . . . . . . 723
3. TheHarmonicModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733
3.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
3.2 Homoclinicpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734
3.3 Symboliccoversoftheharmonicmodel . . . . . . . . . . . . . . . . . 735
3.4 Kernelsofcoveringmaps . . . . . . . . . . . . . . . . . . . . . . . . . 739
4. TheAbelianSandpileModel . . . . . . . . . . . . . . . . . . . . . . . . . 744
5. TheCriticalSandpileModel . . . . . . . . . . . . . . . . . . . . . . . . . . 747
5.1 Surjectivityofthemapsξg: R∞ −→ Xf(d) . . . . . . . . . . . . . . . 747
5.2 Propertiesofthemapsξ , g ∈ I˜ . . . . . . . . . . . . . . . . . . . . . 754
g d
6. TheDissipativeSandpileModel . . . . . . . . . . . . . . . . . . . . . . . . 755
6.1 Thedissipativeharmonicmodel . . . . . . . . . . . . . . . . . . . . . 755
6.2 Thecoveringmapξ(γ): R(∞γ) −→ Xf(d,γ) . . . . . . . . . . . . . . . . 756
7. ConclusionsandFinalRemarks . . . . . . . . . . . . . . . . . . . . . . . . 758
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
722 K.Schmidt,E.Verbitskiy
1. Introduction
Foranyintegerd ≥2let
(cid:2) (cid:4)
(cid:1) (cid:1)
1 1 (cid:3)d
h = ··· log 2d−2 cos(2πx ) dx ···dx , (1.1)
d i 1 d
0 0 i=1
h =1.166,h =1.673,etc.Itturnsoutthatford ≥2,h isthetopologicalentropyof
2 3 d
threedifferentd-dimensionalmodelsinmathematicalphysics,probabilitytheory,and
dynamicalsystems.Ford =2,thereisevenafourthmodelwiththesameentropyh .
d
1.1. Fourmodels. Thed-dimensionalabeliansandpilemodelwasintroducedbyBak,
TangandWiesenfeldin[3,4]andattractedalotofattentionafterthediscoveryofthe
AbelianpropertybyDharin[8].Thesetofinfiniteallowedconfigurationsofthesandpile
model isthe shift-invariantsubset R∞ ⊂ {0,...,2d −1}Zd defined in (4.4) and dis-
cussedinSect.4.1In[10],Dharshowedthatthetopologicalentropyoftheshift-action
σR onR∞ isalsogivenby(3.4),whichimpliesthateveryshift-invariantmeasureµ
∞
of maximal entropy on R∞ has entropy (1.1). Shift-invariant measures on R∞ were
studiedinsomedetailbyAthreyaandJaraiin[1,2],JaraiandRedigin[13];however,
thequestionofuniquenessofthemeasureofmaximalentropyisstillunresolved.
Spanningtreesoffinitegraphsareclassicalobjectsincombinatoricsandgraphthe-
ory.In1991,Pemantleinhisseminalpaper[17]addressedthequestionofconstructing
uniformprobabilitymeasuresonthesetT ofinfinitespanningtreesonZd —i.e.,on
d
the set of spanning subgraphs of Zd without loops. This work was continued in 1993
byBurtonandPemantle[5],wheretheauthorsobservedthatthetopologicalentropyof
thesetofallspanningtreesinZd isalsogivenbytheformula (1.1).Anotherproblem
discussedin[5]istheuniquenessoftheshift-invariantmeasureofmaximalentropyon
T (theproofin[5]isnotcomplete,butSheffieldhasrecentlycompletedtheproofin
d
[22].
This coincidence of entropies raised the question about the relation between these
models.Apartialanswertothisquestionwasgivenin1998byR.Solomyakin[24]:she
constructed injective mappings from the set of rooted spanning trees on finite regions
ofZd into Xf(d) suchthattheimagesaresufficientlyseparated.Inparticular,thispro-
videdadirectproofofcoincidenceofthetopologicalentropiesofαf(d) andσTd without
makinguseofformula(1.1).
Indimension2,spanningtreesarerelatednotonlytothesandpilemodels(cf.e.g.,
[19] for a detailed account) and, by [24], to the harmonic model, but also to a dimer
model(moreprecisely,totheevenshift-actiononthetwo-dimensionaldimermodel)by
[5].
However,theconnectionsbetweentheabeliansandpilesandspanningtrees(aswell
asdimersindimension2),arenon-local:theyareobtainedbyrestrictingthemodelsto
finiteregionsinZd (orZ2)andconstructingmapsbetweentheserestrictions,butthese
mapsarenotconsistentasthefiniteregionsincreasetoZd.
Inthispaperwestudytherelationbetweentheinfiniteabeliansandpilemodelsand
thealgebraicdynamicalsystemscalledtheharmonicmodels.Thepurposeofthispaper
istodefineashift-equivariant,surjectivelocalmappingbetweenthesemodels:fromthe
1 Inthephysicsliteratureitismorecustomarytoviewthesandpilemodelasasubsetof{1,...,d}Zd by
adding1toeachcoordinate.
AbelianSandpilesandtheHarmonicModel 723
infinite critical sandpile model R∞ to the harmonic model. Although we are not able
to prove that this mapping is almost one-to-one it has the property that it sends every
shift-invariantmeasureofmaximalentropyonR∞toHaarmeasureonXf(d).Moreover,
itshedssomelightonthesomewhatelusivegroupstructureofR∞.
Firstly,thedualgroupof Xf(d) isthegroup
G = R /(f(d)),
d d
where R = Z[u±,...,u±] is the ring of Laurent polynomials with integer coeffi-
d 1 d
cientsinthevariablesu ,...,u ,and(f(d))istheprincipalidealin R generatedby
(cid:5) 1 d d
f(d) =2d− d (u +u−1).ThegroupG isthecorrectinfiniteanalogueofthegroups
i=1 i i d
of additionoperatorsdefinedonfinitevolumes,see[9,19](cf.Sect.7).
Secondly,themapξ constructedinthispapergivesrisetoanequivalencerelation∼
Id
onR∞with
x ∼ y ⇐⇒ x −y ∈ker(ξ ),
Id
such that R∞/∼ is a compact abelian group. Moreover, R∞/∼, viewed as a dynami-
calsystemunderthenaturalshift-actionofZd,hasthetopologicalentropy(1.1).This
extends the result of [16], obtained in the case of dissipative sandpile model, to the
criticalsandpilemodel.
Finally,wealsoidentifyanalgebraicdynamicalsystemisomorphictothedissipative
sandpilemodel.Thisallowsaneasyextensionoftheresultsin[16]:namely,theunique-
nessofthemeasureofmaximalentropyonthesetofinfiniterecurrentconfigurations
in the dissipative case. Unfortunately, we are not yet able to establish the analogous
uniquenessresultinthecriticalcase.
1.2. Outlineofthepaper. Sect.2investigatescertainmultipliersofthepotentialfunc-
tion(orGreen’sfunction)ofthesimplerandomwalkonZd.InSect.3theseresultsare
usedtodescribethehomoclinicpointsoftheharmonicmodel.Thesepointsarethenused
todefineshift-equivariantmapsfromthespace(cid:1)∞(Zd,Z)ofallboundedd-parameter
sequencesofintegersto Xf(d).InSect.4weintroducethecriticalanddissipativesand-
pilemodels.InSect.5weshowthatthemapsfoundinSect.3sendthecriticalsandpile
modelR∞ onto Xf(d),preservetopologicalentropy,andmapeverymeasureofmaxi-
malentropyonR∞toHaarmeasureontheharmonicmodel.Afterabriefdiscussionof
furtherpropertiesofthesemapsinSubsect.5.2,weturntodissipativesandpilemodels
in Sect. 6 and define an analogous map to another closed, shift-invariant subgroup of
TZd.Themainresultin[16]showsthatthismapisalmostone-to-one,whichimplies
that the measure of maximal entropy on the dissipative sandpile model is unique and
Bernoulli.
2. APotentialFunctionandits(cid:1)1-Multipliers
Letd ≥1.Foreveryi =1,...,dwewritee(i) =(0,...,0,1,0,...,0)fortheithunit
vectorinZd,andweset0=(0,...,0)∈Zd.
WeidentifythecartesianproductW =RZd withthesetofformalrealpowerseries
d
inthevariablesu±1,...,u±1 byviewingeachw =(w )∈ W asthepowerseries
1 d (cid:3) n d
w un (2.1)
n
n∈Zd
724 K.Schmidt,E.Verbitskiy
with w ∈ R and un = un2···und for every n = (n ,...,n ) ∈ Zd. The involution
w (cid:9)→wn∗onW isdefined1by d 1 d
d
wn∗ =w−n, n∈Zd. (2.2)
For E ⊂ Zd we denote by π : W −→ RE the projection onto the coordinates
E d
in E.
Forevery p ≥1weregard(cid:1)p(Zd)asthesetofallw ∈ W with
d
⎛ ⎞
1/p
(cid:3)
(cid:11)w(cid:11) =⎝ |w |p⎠ <∞.
p n
n∈Zd
Similarlyweview(cid:1)∞(Zd)asthesetofallboundedelementsinW ,equippedwiththe
d
supremumnorm(cid:11)·(cid:11)∞.Finallywedenoteby Rd =Z[u±11,...,u±d1]⊂(cid:1)1(Zd)⊂ Wd
theringofLaurentpolynomials(cid:5)withintegercoefficients.Everyhinanyofthesespaces
willbewrittenash =(hn)= n∈Zd hnun withhn ∈R(resp.hn ∈Zforh ∈ Rd).
Themap(m,w)(cid:9)→um·wwith(um·w)n =wn−misaZd-actionbyautomorphisms
oftheadditivegroupW whichextendslinearlytoan R -actiononW givenby
d d d
(cid:3)
h·w = h un·w (2.3)
n
n∈Zd
foreveryh ∈ R andw ∈ W .Ifwalsoliesin R thisdefinitionisconsistentwiththe
d d d
usualproductin R .
d
For the following discussion we assume that d ≥ 2 and consider the irreducible
Laurentpolynomial
(cid:3)d
f(d) =2d− (u +u−1)∈ R . (2.4)
i i d
i=1
Theequation
f(d)·w =1 (2.5)
with w ∈ W admits a multitude of solutions.2 However, there is a distinguished (or
d
fundamental)solutionw(d)of (2.5)whichhasadeepprobabilisticmeaning:itisacer-
tainmultipleofthelatticeGreen’sfunctionofthesymmetricnearest-neighbourrandom
walkonZd (cf.[6,12,25,27]).
Definition2.1. For every n = (n ,...,n ) ∈ Zd and t = (t ,...,t ) ∈ Td we set
(cid:5) 1 d 1 d
(cid:12)n,t(cid:13)= d n t ∈T.Wedenoteby
j=1 j j
(cid:3) (cid:3)d
F(d)(t)= f(d)e2πi(cid:12)n,t(cid:13) =2d−2· cos(2πt ), t =(t ,...,t )∈Td, (2.6)
n j 1 d
n∈Zd j=1
theFouriertransformof f(d).
2 UndertheobviousembeddingofRd (cid:7)→(cid:1)∞(Zd,Z),theconstantpolynomial1∈Rdcorrespondstothe
elementδ(0)∈(cid:1)∞(Zd,Z)givenby (cid:10)
δ(0)= 1 if n=0,
n 0 otherwise.
AbelianSandpilesandtheHarmonicModel 725
(1)Ford =2,
(cid:1)
e−2πi(cid:12)n,t(cid:13)−1
w(2) := dt forevery n∈Z2.
n T2 F(2)(t)
(2)Ford ≥3,
(cid:1)
e−2πi(cid:12)n,t(cid:13)
w(d) := dt forevery n∈Zd.
n Td F(d)(t)
The difference in these definitions for d = 2 and d > 2 is a consequence of the fact
thatthesimplerandomwalkonZ2 recurrent,whileonhigherdimensionallatticesitis
transient.
Theorem2.2.([6,12,25,27])Wewrite(cid:11)·(cid:11)fortheEuclideannormonZd.
(i)Foreveryd ≥2,w(d)satisfies(2.5).
(ii)Ford =2,
⎧
⎨ 0 if n=0,
wn(2) =⎩− 1 log(cid:11)n(cid:11)−κ −c (cid:11)n1(cid:11)4(n41+n42)−34 +O((cid:11)n(cid:11)−4) if n(cid:14)=0, (2.7)
8π 2 2 (cid:11)n(cid:11)2
whereκ > 0andc > 0.Inparticular,w(2) = 0andw(2) < 0foralln (cid:14)= 0.
2 2 0 n
Moreover,
(cid:3)∞
4·w(2) = (P(X =n|X =0)−P(X =0|X =0)),
n k 0 k 0
k=1
where(X )isthesymmetricnearest-neighbourrandomwalkonZ2.
k
(iii)Ford ≥3,
(cid:5)
1 d n4− 3
(cid:11)n(cid:11)d−2w(d) =κ +c (cid:11)n(cid:11)4 i=1 i d+2 +O((cid:11)n(cid:11)−4) (2.8)
n d d (cid:11)n(cid:11)2
as(cid:11)n(cid:11)→∞,whereκ >0,c >0.Moreover,
d d
(cid:3)∞
2d·w(d) = P(X =n|X =0)>0 forevery n∈Zd,
n k 0
k=0
where(X )isagainthesymmetricnearest-neighbourrandomwalkonZd.
k
Definition2.3.Letw(d) ∈ W bethepointappearinginDefinition2.1.Weset
d
(cid:14) (cid:15)
I = g ∈ R :g·w(d) ∈(cid:1)1(Zd) ⊃(f(d)), (2.9)
d d
where(f(d)) = f(d)· Rd istheprincipalidealgeneratedby f(d).Sincewn(d) = w−(dn)
foreveryn∈Zd itisclearthat I = I∗ ={g∗ :g ∈ I }.
d d d
726 K.Schmidt,E.Verbitskiy
Theorem2.4.Theideal I isoftheform
d
I =(f(d))+I3, (2.10)
d d
where
I ={h ∈ R :h(1)=0}=(1−u )· R +···+(1−u )· R (2.11)
d d 1 d d d
with1=(1,...,1).
FortheproofofTheorem2.4weneedseverallemmas.Weset
J =(f(d))+I3 ⊂ R . (2.12)
d d d
(cid:5)
Lemma2.5.Let g = k∈Zd gkuk ∈ Rd. Then g ∈ Jd if and only if it satisfies the
followingconditions(2.13)–(2.16).
(cid:3)
g =0, (2.13)
k
(cid:3) k∈Zd
g k =0 for i =1,...,d, (2.14)
k i
k=((cid:3)k1,...,kd)∈Zd
g k k =0 for 1≤i (cid:14)= j ≤d, (2.15)
k i j
(cid:3)k=(k1,...,kd)∈Zd
g (k2−k2)=0 for 1≤i (cid:14)= j ≤d. (2.16)
k i j
k=(k1,...,kd)∈Zd
Proof. Condition(2.13)isequivalenttosayingthatg ∈I .Inconjunctionwith(2.13),
d
(2.14)isequivalenttosayingthatg ∈I2:indeed,ifg ∈I ,thenitisoftheform
d d
(cid:3)d
g = (1−u )·a (2.17)
i i
i=1
witha ∈ R fori =1,...,d.Then
i d
∂g = (cid:3) g k ·uk1···ukj−1···ukd =−a +(cid:3)d (1−u )· ∂ai ,
∂u k j 1 j d j i ∂u
j k=(k1,...,kd)∈Zd i=1 j
and ∂g (1)=0ifandonlyifa ∈I .
∂uj j d
Ifg ∈I isoftheform(2.17)andsatisfies(2.14)weset
d
(cid:3)d
aj = (1−ui)·bi,j (2.18)
i=1
withbi,j ∈ Rd.Condition(2.15)issatisfiedifandonlyif
∂2g ∂a ∂a
∂u ∂u (1)=−∂ui − ∂uj =bi,j(1)+bj,i(1)=0
i j j i
for1≤i (cid:14)= j ≤d.
AbelianSandpilesandtheHarmonicModel 727
Finally,if g satisfies(2.13)–(2.14)andisoftheform (2.17)–(2.18)withbi,j ∈ Rd
foralli, j,then(2.16)isequivalenttotheexistenceofaconstantc∈Rwith
(cid:3) ∂a
gkki2 =−2∂ui(1)=2bi,i(1)=c
k=(k1,...,kd)∈Zd i
fori =1,...,d.
The last equation shows that bi,i −b1,1 ∈ Id for i = 2,...,d. By combining all
theseobservationswehaveprovedthatgsatisfies(2.13)–(2.16)ifandonlyifitisofthe
form
(cid:3)d
g =h · (1−u )2+h (2.19)
1 i 2
i=1
with c ∈ Z, h ∈ R and h ∈ I3. The set of all such g ∈ R is an ideal which we
1 d 2 (cid:5)d d
denoteby J˜.Clearly,I3 ⊂ J˜and d (1−u )2 ∈ J˜.Since(1−u )2·(1−u−1)∈I3
d i=1 i i i d
fori =1,...,d aswell,weconcludethat
(cid:3)d (cid:3)d
f(d) = (1−u )2− (1−u−1)·(1−u )2 ∈ J˜. (2.20)
i i i
i=1 i=1
Thisshowsthat J˜⊂ J ,andthereverseinclusionalsofollowsfrom(2.20)and(2.19).
d
(cid:17)(cid:18)
Lemma2.6. I ⊂ J .
d d
Proof. Weassumethatg ∈ Id an(cid:5)dsetv = g·w(d).Inordertoverify(2.13)weargue
bycontradictionandassumethat g (cid:14)=0.Ifd =2then
k k
(cid:5)
g
v =− k k log(cid:11)n(cid:11)+l.o.t.,
n 2π
forlarge(cid:11)n(cid:11).Ifd ≥3,then
(cid:5)
κ g
v = d k k +l.o.t.
n (cid:11)n(cid:11)d−2
forlarge(cid:11)n(cid:11).Inbothcasesitisevidentthatv (cid:14)∈(cid:1)1(Zd).
Bytaking(2.13)intoaccountonegetsthat,foreveryd ≥2,
(cid:3)
v =(g·w(d)) = g w(d)
n n k n−k
(cid:1)k (cid:5)
g e2πi(cid:12)k,t(cid:13)
= e−2πi(cid:12)n,t(cid:13) k(cid:5)k dt.
Td 2d−2 dj=1cos(2πtj)
Hencev =(v )isthesequenceofFouriercoefficientsofthefunction
n
(cid:5)
g e2πi(cid:12)k,t(cid:13)
H(t)= k(cid:5)k .
2d−2 d cos(2πt )
j=1 j
728 K.Schmidt,E.Verbitskiy
Ifv ∈(cid:1)1(Zd),then H mustbeacontinuousfun(cid:5)ctiononTd.Sincet =0istheonly
zeroof F(d) onTd (cf.(2.6)),thenumeratorG = g e2πi(cid:12)k,·(cid:13) mustcompensatefor
k k
thissingularity.ConsidertheTaylorseriesexpansionofG att =0:
(cid:3) (cid:3)d (cid:3) (cid:3)d (cid:3) (cid:3) (cid:3)
G(t)= g +2πi t g k −2π2 t2 g k2−4π2 t t g k k
k j k j j k j i j k i j
k j=1 k j=1 k i(cid:14)=j k
+h.o.t.
TheTaylorseriesexpansionof F(d)att =0isgivenby
(cid:3)d
F(d)(t)=4π2 t2+h.o.t.
j
j=1
Supposethat
(cid:5) (cid:5) (cid:5)
h(t)= a0+ dj=1bjtj + dj=1cjt2j + i(cid:14)=jdi,jtitj +h.o.t
t2+···+t2+h.o.t
1 d
iscontinuousatt =0.Then
a =0, b =0 forall j, c =c forall j, d =0 foralli (cid:14)= j,
0 j j ij
andforsomeconstantc.Ifanyoftheseconditionsisviolated,thenoneeasilyproduces
examplesofsequencest(m) → 0asm → ∞withdistinctlimitslimm→∞h(t(m)).By
applyingthisto H weobtain(2.13)–(2.16),sothatg ∈ J byLemma2.5. (cid:17)(cid:18)
d
ToestablishtheinclusionJ ⊆ I ,wehavetoshowthatforanyg ∈ J ,g·u ∈(cid:1)1(Zd),
d d d
whereu ∈ W oftheform
d
(cid:5)
ω = id=1ni4, or ω = 1 with γ ≥d−2.
n (cid:11)n(cid:11)d+4 n (cid:11)n(cid:11)γ
Ford =2,wealsohavetotreatthecaseω =log(cid:11)n(cid:11).
n
Theseresultsareobtainedinthefollowingthreelemmas.
Lemma2.7.Supposethatd ≥2andthatω∈ W isgivenby
d
(cid:10)
0 if n=0,
ω = (cid:5)
n id=1ni4 if n(cid:14)=0.
(cid:11)n(cid:11)d+4
Ifg ∈ R satisfies(2.13),theng·ω∈(cid:1)1(Zd).
d
Proof. Let M =max{(cid:11)k(cid:11):g (cid:14)=0},andsupposethat(cid:11)n(cid:11)> M.Then
k
(cid:5) (cid:5)
(g·ω) =(cid:3)g id=1(ni −ki)4 =(cid:3)g id=1ni4+O((cid:11)n(cid:11)3)
n k (cid:11)n−k(cid:11)d+4 k(cid:11)n(cid:11)d+4(1+O((cid:11)n(cid:11)−1))
(cid:5)k (cid:2) (cid:4) (cid:16) k (cid:17) (cid:16) (cid:17)
= id=1ni4 (cid:3)g +O 1 =O 1 .
(cid:11)n(cid:11)d+4 k (cid:11)n(cid:11)d+1 (cid:11)n(cid:11)d+1
k
(cid:5)
Therefore, |(g·ω) |<∞. (cid:17)(cid:18)
n n
AbelianSandpilesandtheHarmonicModel 729
For the reverse inclusion J ⊂ I we need different arguments for d = 2 and for
d d
d ≥3.Westartwiththecased =2.
(cid:5)
Lemma2.8.Suppose that g = k∈Z2gkuk ∈ R2 satisfies (2.13). We set S+ =
{k:gk >0}and S− ={k:gk <0}.Put
(cid:3) (cid:3)
M =2 g =2 |g |
g k k
k∈S+ k∈S−
anddefinetwopolynomialsinthevariables(n ,n ):
1 2
(cid:18) (cid:19) (cid:20) (cid:18)
P (n ,n )= (n −k )2+(n −k )2 gk = (cid:11)n−k(cid:11)2gk,
+ 1 2 1 1 2 2
k(cid:18)∈S+ (cid:19) (cid:20) k∈(cid:18)S+ (2.21)
P−(n1,n2)= (n1−k1)2+(n2−k2)2 |gk| = (cid:11)n−k(cid:11)2|gk|.
k∈S− k∈S−
Letmg bethedegreeof P = P+− P−.If
M −m ≥3, (2.22)
g g
theng·ω∈(cid:1)1(Z2),where
(cid:10)
0 if n=(0,0),
ω =
n log(cid:11)n(cid:11) if n(cid:14)=(0,0).
(cid:5)
Proof. Since k∈Z2gk =0by(2.13), Mg =degP+ =degP−and
mg =degP <max(degP+,degP−)= Mg.
Letv =g·ω.Hence,forallnwith(cid:11)n(cid:11)>max{(cid:11)k(cid:11):k∈ S+∪S−},onehas
(cid:21) (cid:21) (cid:21) (cid:16) (cid:17)(cid:21)
|(g·ω)n|= 21(cid:21)(cid:21)(cid:21)log PP−+((nn11,,nn22))(cid:21)(cid:21)(cid:21)= 21(cid:21)(cid:21)(cid:21)log 1+ P+(n1,Pn−2)(n−1,Pn−2()n1,n2) (cid:21)(cid:21)(cid:21).
ThereexistconstantsC,N suchthat
(cid:21) (cid:21)
(cid:21)(cid:21)(cid:21)P+(n1,n2)− P−(n1,n2)(cid:21)(cid:21)(cid:21)≤C(cid:11)n(cid:11)mg = C < 1
P−(n1,n2) (cid:11)n(cid:11)Mg (cid:11)n(cid:11)Mg−mg 2
for(cid:11)n(cid:11)≥ N.HencewecanfindanotherconstantC˜ suchthat
˜
C
|(g·ω) |≤
n (cid:11)n(cid:11)Mg−mg
forallsufficientlylarge(cid:11)n(cid:11).SinceM −m ≥3,wefinallyconcludethatg·ω∈(cid:1)1(Z2).
g g
(cid:17)(cid:18)
Lemma2.9.Supposethatg ∈ J (cf.(2.13)–(2.16)),andthatω∈ W isgivenby
d d
(cid:10)
0 if n=0,
ω =
n 1 if n(cid:14)=0,
(cid:11)n(cid:11)γ
forsomeintegerγ ≥d−2.Theng·ω∈(cid:1)1(Zd).