A general renormalization procedure on the one-dimensional lattice and decay of correlations

0.13 MB·

by  Artur O. Lopes

A GENERAL RENORMALIZATION PROCEDURE ON THE ONE-DIMENSIONAL LATTICE
AND DECAY OF CORRELATIONS
A.O.LOPES
7
1
0
Inst. deMatema´tica,UFRGS-PortoAlegre,Brasil
2

n
a
J

6
2 ABSTRACT. WepresentageneralformofRenormalizationoperatorRactingonpotentialsV: 0,1 N R.
WeexhibittheanalyticalexpressionofthefixedpointpotentialV forsuchoperatorR. Thispo{tentia}lc→anbe
] expressedinanaturallywayintermsofacertainintegralovertheHausdorffprobabilityonaCantortypeset
S
ontheinterval[0,1]. ThisresultgeneralizesapreviousonebyA.Baraviera,R.LeplaideurandA.Lopes(see
D
[2])wherethefixedpointpotentialV wasofHofbauertype(see[16]).
h. ForthepotentialsofHofbauertype(awellknowncaseofphasetransition)thedecayisliken−g,g >0(see
[22][23],[14]or[10])
t
a Amongotherthingswepresenttheestimationofthedecayofcorrelation oftheequilibrium probability
m associatedtothefixedpotentialVofourgeneralrenormalizationprocedure.Insomecaseswegetpolynomial
[ decaTyhleikpeotne−ngti,agls>g0w,eancdonisnidoethrehresreaadreecaeylefmasetnetrstohfanthnees−o√canl,lwedhfeanmnil→yo¥f.Walterspotentialson 0,1 N(see
{ }
1 [27])whichgeneralizes thepotentials consideredinitiallybyF.Hofbauerin[16]. Forthesepotentialssome
v explicitexpressionsfortheeigenfunctionsareknown.
6 Inafinalsectionwealsoshowthatgivenanychoicedn 0ofrealnumbersvaryingwithn Nthere
→ ∈
5 existapotentialgontheclassdefinedbyWalterswhichhasainvariantprobabilitywithsuchnumbersasthe
6 coefficientsofcorrelation(foracertainexplicitobservablefunction).
7
0 1. INTRODUCTION
.
1
ItisaclassicalprobleminStatisticalMechanicsthestudyofphasetransitions(see[12],[13],[1],[15],
0
7 [28]and[29])forgeneralpotentials. Inseveralexamplesitisknownthatonthepointofphasetransition
1 one gets polynomialdecay of correlationfor the associated equilibriumprobability. As far as we know,
: thereisnounifiedwayforunderstandingallkindsofphasetransitions.
v
i One important technique on the analysis of such class of problems is the use of the renormalization
X
operator (see for instance [5], [18], [4] and [8]). The potential which is fixed for the renormalization
r operatorinmostofthecasesisquiterelevantfortheunderstandingoftheproblem.
a
PhasetransitionsinthesettingofThermodynamicFormalismhasbeenconsideredfromdifferentpoints
ofview(seeforinstance[2],[14],[19],[20],[21],[7],[6],[22],[23],[9],[10],[11]and[17]).
In[2]itwasintroducedarenormalizationoperator(actingonpotentials)ontheone-dimensionallattice
N
0,1 andthefixedpointonegetsisthewellknownHofbauerpotential. Theequilibriumprobabilityfor
{ }
suchpotentialexhibitpolynomialdecayofcorrelationforacertainclassofobservables.
Here we consider a more general procedure of renormalization of potentials defined on the lattice
N
0,1 . We want to exhibit a fixed point potential for such procedure. The fixed point potential will
{ }
beonaclassoffunctionspreviouslyconsideredbyP.Waltersin[27](whichincludestheHofbauerpoten-
tial). Weareparticularyinterestedinthedecayofcorrelation(foracertainnaturalfunction)associatedto
theequilibriumprobabilityforsuchpotential.
In some cases (first type) the fixed point will produce polynomialdecay of correlation and in others
cases(secondtype)notofpolynomialtype(someofthemfasterthanne √n,whenn ¥ ).
−
→
Inordertogiveanideatothereaderofthekindofresultswewillgetherewepresentanexample: in
aparticularcasetherenormalizationoperatorR actsoncontinuouspotentialsU, suchthatU isconstant
Date:January27,2017.
1
2 A.O.LOPES
oncylindersetsoftheform0...01and1...10,forallQ N.ForeachfixedQthevaluesarethesameon
∈
Q Q
bothcylindersets
WedefineU =R(U )in|t{hze}followi|ng{zw}ay: supposexisoftheformx=(0...0,1,...),then
2 1
Q
U (x) = R(U )(x)=U (0...0,1)+U (0...0,1).|{z}
2 1 1 1
3Q 3Q 2
−
InthiscasethefixedpointpotentialV fortherenor|m{azli}zationoper|a{tozr}R isthefunction
1
V(x)= dn (t)
− K (Q t)a
Z −
when x=(0...0,1,...), Q N, and where n is the Hausdorff probability on the classical Cantor set (a
∈
Q
subsetoftheinterval[0,1])andwhichhasHausdorffdimensiona = log2.
|{z} log3
Inseveraldistinctclassesofproblemsthefixedpointpotentialfortherenormalizationoperatorcanbe
expressed on an integral form associated to a fractal probability (see for instance [5], [3], [18], [4] and
[24]).Herewewillalsogetasimilarkindofexpression.
We will present now the main definitions giving more details on the specific problems we want to
analyze.
We will consider a subclass of potentials g: 0,1 N R which were described in [27]. Using the
notationofthatpaper,considerarealsequencea{, n } N→, andassumethata 0, whenn ¥ . Inour
n n
∈ → →
modela=0=c.
Givena ,a ,...,a ,...<0defineh , j 1,insuchwaythat(a +a +...+a )=logh , j 1.
2 3 n j 2 3 j+1 j+1
Forthesequence1 h >0,n N,n≥ 1,h =1,weassumethat,h 0,and(cid:229) ¥ h g =W(≥b )>1
foranyb >1. ≥ n ∈ ≥ 1 n→ n=1 n
log2
Particularcaseswhichweareinterestedarewhenh =e n1/2,h =e n1−log5 orh =n g,g >2.
n − n − n −
Considerthetwoinfinitecollectionsofcylindersetsgivenby
L =000...01 and R =111...10, foralln 1.
n n
≥
n n
Usingtheseparametersweca|n{dzefi}neacontinuous|po{teznt}ialg=gb :W R, whichissimplydenote
byg,inthefollowingway:foranyx W andb 1 →
∈ ≥
b a , ifx L , forsomen 2;
n n
∈ ≥
b an, ifx Rn, forsomen 2;
g(x)=−llooggWW((bb )),, iiffxx∈∈LR1;; ≥
1
− ∈
¥ ¥
Intheabovewearetakingcn=an0,,foralln. ifx∈{1 ,0 }.
WearemainlyinterestedinpotentialsgwhicharenotofHo¨ldertype.
Definition 1. Given A:W = 0,1 N R, the Ruelle operator L acts on functions y :W R in the
A
{ } → →
followingway
d
j (x)=L (y )(x)= (cid:229) eA(ax)y (ax).
A
a=1
BythiswemeanL (y )=j .
A
From the generalresult described in page 1341in [27] applied to our particular case, we get that the
eigenfunctionj b oftheRuelleoperatorforA=b g,forb 1,issuchthat,foranyn 1
≤ ≥
AGENERALRENORMALIZATIONPROCEDUREONTHEONE-DIMENSIONALLATTICEANDDECAYOFCORRELATIONS 3
¥ h b
j b (0n1...)=a 1+h1nb j(cid:229)=1l (nb+)jj!, (1)
¥ h b
j b (1n0...)=b(b ) 1+h1nb j(cid:229)=1l (nb+)jj!, (2)
j b (0¥ )=a and j b (1¥ )=b(b ).
Theeigenfunctionisunique.
Weassumethatforb =1wegetl (b )=1=a =b(b )
Whenb =1,itiseasytoseethatifitiswelldefinedtheprobabilityr ontheBoreliansofW ,suchthat,
foranynaturalnumberq 1,wehavethat
≥
r 0q1 =h and r 1q0 =h . (3)
q q
then,r isaneigenprobabilityforthedualoftheRuelleoperatorforg.
(cid:0) (cid:1) (cid:0) (cid:1)
Therefore,fromageneralprocedure(see[25])onegetsthattheprobabilitym ,suchthat,
¥ ¥
m 0q1 =h j (0n1...)= (cid:229) h and m 1q0 =h j (1n0...)= (cid:229) h (4)
q n+j q n+j
j=0 j=0
(cid:0) (cid:1) (cid:0) (cid:1)
istheequilibriumprobabilityforg.
We will show in Theorem 6 that decay of correlation d , as a function of q N, of the equilibrium
q
∈
probabilitym forthepotentialgandfortheindicatorofthecylinder0isoforder
¥ ¥
(cid:229) (cid:229) h .
(k+q+s)
s=1k=0
Given a certain sequence d 0, there exists (under mild conditions) a choice of h such that d =
q n q
(cid:229) ¥ (cid:229) ¥ h . Indeed, firs→t for the given sequence d , q N, take inductively c such that c =
ds=1d k=0,n(k+Nq+.sI)nthisway(cid:229) ¥ c =d . q ∈ n n
n−Nown+,1give∈nsuchsequencecj,=nq jN,taqkeanewsequenceh ,suchthat,h =c c ,r N.Inthis
n r r r r+1
case(cid:229) ¥ h =c . ∈ − ∈
i=p i p
Itfollowsthatforallqwegetd =(cid:229) ¥ (cid:229) ¥ h .
q s=1 k=0 (k+q+s)
Partoftheresultspresentedherewerea consequenceoffruitfuldiscussionswithA. BaravieraandR.
Leplaideurwhichcontributedinanontrivialwaytoourreasoning.Ourthankstothem.
Insections2and3wepresentresultsaboutthefixedpointpotentialforacertainrenormalizationopera-
tor. Onsections4and5weestimatethedecayofcorrelationsfortheclassopotentialsofWalterstypeand
weshowthatalargeclassoftypesofdecayofcorrelationsareattainedbysuchfamily. Ontheappendix
weuseresultsoftheprevioussectionstoshowthatforsomeofthesefixedpointpotentialsthedecayisnot
ofpurelypolynomialtype.
2. A GENERAL RENORMALIZATION- FIRST TYPE
Thepotentials f whichweareinterestedareontheclass
f(0n1z)=a , f(0¥ )=a, f(10n1z)=d , f(10¥ )=d,
n n
f(01n0z)=b , f(01¥ )=b, f(1n0z)=c , f(1¥ )=c,
n n
andisassumedthata a,b b,c candd d.
n n n n
→ → → →
Wewillassumethatb =d =b=d.
n n
Therenormalizationoperatorwillactonsuchsetoffunctions.
Fixanaturalnumberk 2.Wewillgeneralizethekindofrenormalizationdescribedin[2]wherek=2.
≥
Nowtakeforn>1
H (0n1x)=0kn (k 2)1x
k − −
H (1n0y)=1kn (k 2)0y
k − −
and
H (01mx)=01mx
k
4 A.O.LOPES
H (10my)=10my
k
¥ ¥ ¥ ¥
andalsoH (0 )=0 ,H (1 )=1 .
k k
Nowdefinetherenormalizationoperatoras
Rk(V)(x)=V(Hk(x))+V(s Hk(x))+V(s 2Hk(x))+...+V(s k−1Hk(x)).
WewillshowthatitispossibletoexhibitafixedpointV fortherenormalizationoperatorR .
k
Theclassoffunctionsweconsiderhereinthesection1islargeenoughtobeabletogetfixedpointsV.
Thismeanswewillbeabletofindasequencea ,n N,whichwilldefinesuchV
n
∈
An interesting question is to estimate the decay of correlation of the equilibrium probability for the
fixedpointpotentialV. Onecanshow(seesection5)thatforthecaseofrenormalizationoffirsttypethe
potentialV issuchthatwegetpolynomialdecayofcorrelation.
¥ ¥
IfV isfixedfortherenormalizationoperatorR ,thenV(1 )=V(0 )=0=a=c.
k
Moreover,
¥ ¥ ¥ ¥
V(01 )=V(01 )+V(1 )+V(1 )=b
and
¥ ¥ ¥ ¥
a =V(001 )=V(00...0001 )+V(00...0001 )+...+V(0001 )=a +a +...+a .
2 3 4 k+2
k+2 k+1
| {z } | {z }
¥ ¥ ¥ ¥
a =V(0001 )=V(00...0001 )+V(00...0001 )+...+V(00...0001 )=a +a +...+a .
3 k+3 k+4 2k+2
2k+2 2k+1 k+3
Aswewillseoncew|efi{zxth}evalueof|a2{azdb}thepotentialV|w{ilzlb}edetermined.
TheWalterspotentialV whichisthefixedpointforR shouldsatisfyforn 2:
k
≥
a =a +a +...+a
n k(n 1)+2 k(n 1)+1 k(n 2)+3
− − −
a =c =c +c +...+c
n n k(n 1)+2 k(n 1)+1 k(n 2)+3
− − −
and
d =b =b=d
n n
areconstant.
Nowwesolvea (2)asthesolutionof
2+a (2)
a = log .
2 − 2+a (2) 1
−
Now,a (n)isdefinedbyinductionforn 2,
≥
a =a =...=a ,
k(n 1)+2 k(n 1)+1 k(n 2)+3
− − −
and
a =ka (n)+(k 2).
k(n 1)+2
− −
Takinga =d ,n 2,oftheform
n n
≥
n+a (n)
a = log <0
n − n+a (n) 1
−
onecanshowthatwegetthefixedpointsolutionoftheRenormalizationoperator.
Wepointoutthata andbarefree.
2
OnecanshowthatinthiscaseforsuchV thereexistsg suchthatlogh g logn,n 1.Ifg >2,one
n
getspolynomialdecayofcorrelationoftheformn2 g similartotheHofba∼ue−rcase. This≥kindofquestion
−
waspreviouslyconsiderin[22][23][14][10]. Thiscanbealsoobtainedfromthegeneralproofofsection
5.
AGENERALRENORMALIZATIONPROCEDUREONTHEONE-DIMENSIONALLATTICEANDDECAYOFCORRELATIONS 5
3. A GENERALRENORMALIZATION -SECOND TYPE
We consider in this section a far more general generalization of the renormalization operator and we
exhibitthefixedpointpotentialV.
Wepointoutthatinmanyexamplesthefixedpointpotentialofarenormalizationoperatorisdescribed
byan expressionobtainedbythe integralofa kernelwith respectto a certainfractalprobability(see for
instance[5],[3],[18],[4]and[24]). Herewewillgetasimilarkindofresult.
Let us denote L the cylinders[00...01] (with n zeros) and R the cylinders[11...10] (with n ones).
n n
Letsusdefineapotentialthatisa onL andb onR .
n n n n
Inordertodefinethesecondtyperenormalizationoperator,wefirstintroducethemapH. Letk bean
integerlargerorequalthan2;takeh(0)=00...0andh(1)=11...1;hencewedefine
k k
H(x1,x2,x|3,{.z..)}=(h(x1)h(x2|)h{(zx3})...).
Itiseasytoseethats kH=Hs . Hence
H(00...01...)=00...01... say,ifx L thenH(x) L
n kn
∈ ∈
n kn
and
| {z } | {z }
H(11...10...)=11...10... say,ifx R thenH(x) R
n kn
∈ ∈
n kn
Wenowdefinetherenormalizationoperatorasthemaponfunctionsdefinedasfollows: fixl suchthat
| {z } | {z }
2 l kandtheconstants0 c <c <...<c k. Then
1 2 l
≤ ≤ ≤ ≤
RV(x):=V(s c1H(x))+V(s c2H(x))+ +V(s clH(x))
···
OnecanseethattheRenormalizationoperatoroflastsection(thecasel=k)isaparticularcaseofthis
newone.
TheequationaboveforthepotentialVcanberewritten,analogouslytotheexpressionsintheprevious
section,as
Ra =a +a + +a
n kn−c1 kn−c2 ··· kn−cl
(andasimilarexpressionholdstoc ).
n
WeareinterestedonfindingafixedpotentialV forsuchrenormalizationoperator.Forapotentialonthe
classofWaltersthefixedpointequationis
a =a +a + +a
n kn−c1 kn−c2 ··· kn−cl
Thenwegetthefollowingresult:
Lemma2. GivenN 1then
≥ RNV(x)=(cid:229) V(s jHN(x))
j
where jisoftheform
j=b k0+b k1+b k2+ +b kN 1
0 1 2 N 1 −
··· −
forb chosenontheset c ,c ,...,c .
i 1 2 l
{ }
Proof. Indeed,theexpressionitobviouslyvalidforN =1(bydefinitionoftheoperatorR). Now,admit
thatitholdsforN;weneedtoshowthatthisimpliestheexpressionforRN+1V. But
RN+1V =R(RNV)=RNs c1H+RNs c2H+ +RNs clH=
···
(cid:229) V(s jHNs c1H)+...
j
Nowitiseasytosee(alsobyinduction)that
Hs PH =s PkH2
and
HPs H=s kPH2(?)= s kPHP+1
6 A.O.LOPES
Withthistwoexpressionsweget
HNs cH=s ckNHN+1
ThenwecanshowthatRN+1V is
(cid:229) V(s jHNs c1H)+...=
j
(cid:229) V(s js c1kNHN+1)+...=(cid:229) V(s j+c1kNHN+1)+...
j j
where j=b k0+b k1+b k2+ +b kN 1;hencetheexpressionaboveisindeed
0 1 2 N 1 −
··· −
(cid:229) V(s JHN+1)+...
J
whereJ=d k0+d k1+d k2+ +d kN 1+d kN,showingthatRN+1 isasclaimed. (cid:3)
0 1 2 N 1 − N
··· −
NowwewanttoshowthatwehaveafixedpointofR thatintheneighborhoodof0¥ behaveslike
1
V(00...01...)
∼− logl
n nlogk
Aswewillseeinamomentitwillbenaturaltoconsiderthe(Cantorlike)setK = K(l,k)whichisthe
| {z }
closureoftheset
x= (cid:229) aik−i forai c1,...,cl ,n Z
{ ∈{ } ∈ }
n 1
≥
Notethatwhenk=lthesetKwillbetheinterval[0,1].
Example3. Considerthefollowingparticularcase.
DefineH :W = 0,1 N W by:
3
{ } →
H ((0,...,0,1,...,10,...,0,1,...))=(0,...,0,1,...,10,...,0,1,...),
3
c1 c2 c3 3c1 c2 c3
and | {z } | {z }| {z } | {z } | {z }| {z }
H ((1,...,1,0,...,01,...,1,1,...))=(1,...,1,0,...,01,...,1,1,...).
3
c1 c2 c3 3c1 c2 c3
Inthiscasetherenormalizationoperatoris
| {z } | {z }| {z } | {z } | {z }| {z }
R (V)(x)=V(H (x))+V(s H (x))+V(s 2H (x))
3 3 3 3
Ifx=(0,...,0,1,..)orx=(1,...,1,0,..)denoteV(x)=log c1 .Notethatforeachc wegetthat
c1−1 1
c1 c1
3c 3c 1 3c 2 c
| {z } | {z1 } 1 1 1
log +log − +log − =log .
3c 1 3c 2 3c 3 c 1
1 1 1 1
− − − −
InthiscaseV isafixedpointfortherenormalizationoperator.
WecanchoosethesignalofV andthen,wetakeV intheform
1 1 c
1
V(1,...,1,0,..) =V(0,...,0,1,..)= dt= log
c1 c1 −Z0 c1−t − c1−1
| {z } | {z }
Inordertofindagoodguess(inthegeneralcase)forthefixedpointpotentialconsider
1
U(x)= , foranyQ
Qa
when,x=0,...,0,1,..),or,x=1,...,1,0,..).
Q Q
Then,byiterationofU weget:
| {z } | {z }
RNU(x)=(cid:229) 1 =(cid:229) 1 1
(QkN j)a kNa (Q j )a
j − j −kN
AGENERALRENORMALIZATIONPROCEDUREONTHEONE-DIMENSIONALLATTICEANDDECAYOFCORRELATIONS 7
for j=b k0+b k1+b k2+ +b kN 1 (andb c ,...,c ).
0 1 2 N 1 − i 1 l
Fromtheaboveseemsnatu··r·altotr−ytofindafixed∈p{ointV oft}heform
1
V(x)= dn (t),
− K (Q t)a
Z −
when(0...0,1,..),forsomeprobabilityn ontheinterval.
Q
Examp|le{z4.}Letusfixk=3,l=2andc =0,c =2. InthiscasetheCantorsetK wegenerateisindeed
1 2
theclassicalCantorset. WecanwriteK=K K whereK [0,1/3]andK [2/3,1].
0 2 0 2
∪ ⊂ ⊂
NotethatthetransformationT(x)=3x(mod1)istwotoonewhenrestrictedtoK.
Moreover,T[0,1/3]=[0,1]=T[2/3,1].
Letn bethemeasuresupportedonK andconsidera =log2/log3,thatistheHausdorffdimensionof
K.
Note that the Radon-Nykodin derivative of n in each injective branch of T restricted to [0,1/3] and
[2/3,1]isequalto2.
Wewanttoshowthatthefunction
1
V(0...01)= dn (t)
− K (Q t)a
Q Z −
isafixedpointoftherenormalizationo|p{ezra}tor,i.e.,
V(x)=V(H(x))+V(s 2H(x))
Ifx=0...01thiscorrespondstoshowthat
Q
|{z} 1 1 1
dn (t)= dn (t)+ dn (t)
K (Q t)a K (3Q t)a K (3Q 2 t)a
Z − Z − Z − −
Notethatthesecondexpressionis
1 1 1
dn (t)+ dn (t) =
3a K (Q t/3)a K (Q 2/3 t/3)a
(cid:18)Z − Z − − (cid:19)
1 1 1
dn (t)+ dn (t)
2 K (Q t/3)a K (Q 2/3 t/3)a
(cid:18)Z − Z − − (cid:19)
Weremindthat 32a =1.
Inthefirstintegralwecanusethechangeofvariablesx=t/3,thenweget
1 1
dn (t)= 2dn (x)
ZK (Q−t/3)a ZK0 (Q−x)a
Inthesecondweusex=2/3+t/3andweget
1 1
dn (t)= 2dn (x).
ZK (Q−2/3−t/3)a ZK2 (Q−x)a
Therefore,
1 1 1
dn (t)+ dn (t) =
3a K (Q t/3)a K (Q 2/3 t/3)a
(cid:18)Z − Z − − (cid:19)
1 1 1
2 dn (x)+ dn (x) =
3a (cid:18)ZK0 (Q−x)a ZK2 (Q−x)a (cid:19)
1 1
dn (x)= dn (x)
ZK0∪K2 (Q−x)a ZK (Q−x)a
asclaimed.
8 A.O.LOPES
Theorem5. Letxoftheform: x=00...01...,orx=11...10...,thentheoperator
Q Q
RV(x):=V|(s{zc1H}(x))+V(s c2|H{(zx))}+ +V(s clH(x))
···
hasafixedpointV oftheform
1
V(x)= dn (t)
− K (Q t)a
Z −
where a = logl, and where n is the measure over K that maximizes the entropy for the transformation
logk
T(x)=kx(mod1)actingonK .
Proof. Thereasoningissimilartothelastexample.Considera = logl.
logk
ThetransformationT(x)=kx(mod1)actingonK isl toone. Themaximalentropyprobabilityn has
entropylogl.
There exists l sets K1,K2,...,Kl [0,1] such that T(Kj K)=K, and by consideringT restricted to
Kj=[c /k,(c +1)/k], j=1,2,...,l⊂,wegetaninjectivemap∩.Considertheinducedprobabilityn =T (n )
j j j ∗
onK . TheRadon-Nykodinderivativeofn withrespectton ineachsetKj isequaltol.
j j
Ifx=(0...0,1,...)(or,x=(1...1,0,...))wewanttoshowthat
Q Q
|{z}V(x)= |1{z}dn (t)= (cid:229) l 1 dn (t)=RV(x).
−ZK (Q−t)a − j=1ZK ((kQ−cj)−t)a
Foreach jwehavethat
1 1 1
dn (t)= dn (t)= dn (t).
ZK ((kQ−cj)−t)a ZK ka (Q−cjk+t)a ZK l(Q−ckj −kt )a
Foreach jweconsiderthechangeofcoordinatet t/konthesetKj.
→
Then,wegetthat
1 1
dn (t)= ldn (y).
K (Q cj+t)a Kj (Q y)a
Z − k Z −
As
1 dn (t)= (cid:229) l 1 dn (t)
ZK (Q−t)a j=1ZKj (Q−t)a
wegettheclaim.
(cid:3)
Notethat
1
dn (t) Q a
− K (Q t)a ∼− −
Z −
forlargeQ. Inthiscaseh n willbeofordere−n1−a .Whenk=5andl=2wewillgetthattheequilibrium
stateforfixedpointV hasadecayfasterthanne √n (seeAppendix).
−
4. THEASSOCIATED JACOBIAN
Nowweareinterestedontheestimationofthedecayofcorrelationassociatedtoequilibriumprobabili-
tiesassociatedtothefixedpointpotentialswegetviatherenormalizationoperator. Wewillshowthatitis
notalwaysofpolynomialtype. Infact,wewillpresentthedecayofcorrelationforageneralpotentialof
Walterstype.
Supposeg is of Walter type. If we add a constantto a potentialits equilibriumstate will notchange.
Then,wecanassumewithoutlostofgeneralitythatthepotentialghaspressurezero.
Theassociated JacobianJ forthe potentialdefinedby g atthe inversetemperatureb =1 is suchthat
logJ=g+logj log(j s ),wherej istheeigenfunctionoftheRuelleoperatorforg(see(2).
− ◦
Wewilldefiner(q)forq 1.
≥
WesplittheexpressionoflogJ(x)insixcases:
AGENERALRENORMALIZATIONPROCEDUREONTHEONE-DIMENSIONALLATTICEANDDECAYOFCORRELATIONS 9
a) forq 2andx L orx R wehave
q q
≥ ∈ ∈
¥
logJ(x)=b a +log 1+h (cid:229) h
q q (n+q 1)
n=2 − !
¥
log 1+h (cid:229) h
(q 1) (n+q 2)
− − n=2 − !
:=b a +logr(q) logr(q 1).
k
− −
q
b) forx=0111...10... L wehave
1
∈
¥
z }| { logJ(x)= log 1+h (cid:229) h = log(r(q)).
q (n+q 1)
− n=2 − ! −
q
c) forx=1000...01... R wehave
1
∈
¥
z }| { logJ(x)= log 1+h (cid:229) h = log(r(q)).
q (n+q 1)
− n=2 − ! −
¥ ¥
d) forx=0 orx=1 wehaveJ(x)=1.
¥ ¥
e) forx=10 orx=01 wehaveJ(x)=0.
Notethatr(1)h =W.
1
Theequilibriumprobabilitym forghasJacobianJand
Ll∗ogJ(m )=m .
5. DECAY OF CORRELATION
In this section we want to estimate the decay of correlation of the observable I for the equilibrium
[0]
probabilitym atthecriticalinversetemperatureT whereb =1= 1.
T
Weassumefromnowonthatb =1.
Thatis,wewillestimatetheasymptoticbehaviorwithkofthedecayofcorrelationoftheobservableI
0
(I s q)[I m [0]]dm a (q),
[0] [0] 1
W ◦ − ∼
Z
wherem istheequilibriumprobabilityforg,whenb =1.
Wewillshow,asaparticularcase,thatifh q g,withg >2,then,a (q)goestozerolikeq2 g. The
q − −
∼
nextresultisquitegeneral.
Theorem6. Thedecayofcorrelationasafunctionofqfortheequilibriumprobabilitym (forthepotential
g)fortheindicatorofthecylinder0is
¥ ¥
(cid:229) (cid:229) h
(k+q+s)
s=1k=0
inthecaseiswelldefined.
Proof:Thetechniqueissimilartotheoneemployedin[14]and[10].
Firstweneedtoestimatetheasymptoticlimit
m [0] Lq (I )(01..).
1 − logJ [0]
Weclaimthat
q ¥
m [0] Lq (I )(01..) (cid:229) (cid:229) h . (5)
1 − logJ [0] ∼ j
n=1 j=q+1
Wewillpresenttheproofofthisfactlater.
10 A.O.LOPES
Now,forafixedqandanys 2weneedtoevaluatethedifference
≥
m [0] Lq (I )(00...01..).
1 − logJ [0]
s
s |{z}
For fixed q and variable s, let Bs =Lq (I )(00...01..), A(t)=Lt (I )(01...) and finally denote
q logJ [0] logJ [0]
psn= r(sr+(sn))hhss+n,n=1,..,q−2,s∈N. z}|{
Wedenotea (q,s)= h (s+q)r(s+q).
h sr(s)
Onecanshow(seefigure3page1092andexpressioninthebottomofpage1090[14]whereasimilar
caseisdescribed)thatforanys,q 2
≥
r(s+1) r(s+2) r(s+q 2)
Bs =A +eas+1 A +eas+1+as+2 A +...+eas+1+as+2+...+aq+s 2 − A +
q q−1 r(s) q−2 r(s) q−3 − r(s) 1
r(s+q)
eas+1+as+2+...+aq+s 1+aq+s =
− r(s)
Aq−1+r(s+r(1s))hh s(s+1)Aq−2+ r(sr+(s2))hhss+2Aq−3+...+r(s+qr−(s2))hhs(s+q−2)A1+r(s+r(qs))hh (ss+q) =
Aq−1+r(s+r(1s))hh s(s+1)Aq−2+ r(sr+(s2))hhss+2Aq−3+...+r(s+qr−(s2))hhs(s+q−2)A1+a (n,s). (6)
Thisisnotakindofrenewalequation.
Letusnowconsiderforn 1fixed,ands 2thesequenceVs=m [0] Bs.
Wealsointroduce,forn ≥1,thesequence≥sV =m [0] A ,annd 1 − n
n 1 n
≥ −
Uns=−m [0](r(s+r(1s))hh s(s+1) + r(sr+(s2))hhss+2+...+r(s+qr−(s2))hhs(s+q−2))−a (n,s).
Fromtheequation(6)wededucethatforqfixed
m [0] Lq I (00...01..) =m [0] Bs =Vs=V +V ps+V ps+...+V ps +Us. (7)
1 − logJ [0] − q q q−1 q−2 1 q−3 2 1 q−2 q
s
Rememberthat(4)|c{lazim}s
m 0s1 =h j (0s1...)=h r(s). (8)
s 1 s
Foreachfixedqwewillhavetoest(cid:0)imat(cid:1)elaterforeachsthevalue
m (0s1) Lq I (00...01..) m [0] = h r(s) Lq I (00...01..) m [0] =
logJ [0] − s logJ [0] −
s s
(cid:2) (cid:3) (cid:2) (cid:3)
|{z} |{z}
V h r(s)+V r(1+s)h +V r(2+s)h +...+V r(s+q 2)h r(s+q)h . (9)
q 1 s q 2 1+s q 3 2+s 1 s+q 2 s+q
− − − − − −
AsmentionedbeforetheRuelleoperatorL isthedual(intheL2(W ,B,m )sense)oftheKoopman
logJ 1
operatorK (j )=j s ,usingthisduality,expressions(9)and(15)wegetthatforfixedqthat
◦