Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus

0.47 MB·English

by  Alexi Morin-Duchesne

Critical exponents for the homology
of Fortuin-Kasteleyn clusters on a torus
9 Alexi Morin-Duchesne∗,
0
De´partementdephysique
0
2 Universite´ deMontre´al,C.P.6128,succ.centre-ville,Montre´al

n Que´bec,Canada,H3C3J7
a
J Yvan Saint-Aubin†

1
De´partementdemathe´matiquesetdestatistique
2
Universite´ deMontre´al,C.P.6128,succ.centre-ville,Montre´al

]
Que´bec,Canada,H3C3J7
h
c
e
m January21,2009
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t
a
t
s
. Abstract
t
a
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AFortuin-Kasteleynclusteronatorusissaidtobeoftype{a,b},a,b∈Z,ifitpossibletodrawacurve
d- belongingtotheclusterthatwindsatimesaroundthefirstcycleofthetorusasitwinds−btimesaround
n thesecond. EventhoughtheQ-Pottsm√odelsmakesenseonlyforQintegers,theycanbeincludedintoa
o familyofmodelsparametrizedbyβ= QforwhichtheFortuin-Kasteleynclusterscanbedefinedforany
c realβ ∈ (0,2]. Forthisfamily,westudytheprobabilityπ({a,b})ofagiventypeofclustersasafunction
[ of the torus modular parameter τ = τr +iτi. We compute the asymptotic behavior of some of these
probabilitiesasthetorusbecomesinfinitelythin. Forexample,thebehaviorofπ({1,0})isstudiedalong
2
v thelineτr =0andτi →∞.Exponentsdescribingthesebehaviorsaredefinedandrelatedtoweightshr,s
5 oftheextendedKactableforr,sintegers,butalsohalf-integers.Numericalsimulationsarealsopresented.
2 Possiblerelationshipwithrecentworksandconformalloopensemblesisdiscussed.
9 Keywords:Fortuin-Kasteleynclusters,torus,homologyprobabilities,homotopyprobabilities,percolation,
2 Isingmodel,logarithmicminimalmodels,SLE.
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∗[email protected]
†[email protected]
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Contents
1 Introduction 2
2 Theprobabilityπ({1,0})inthelimitτ =0,τ →∞ 5
r i
3 Probabilitiesπ({a,b})inthelimitτ =c/d,τ →0 9
r i
4 ThelimitQ→0 11
5 MonteCarlosimulations 13
5.1 ModelswithrationalandirrationalQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Behaviorofπτ({1,2})closetoτ= 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1 Introduction
One of the main observables of two-dimensional percolation is the crossing probability between two dis-
jointsubsetsoftheboundaryofadomain. Thisdomainisusuallytakenhomeomorphictoadisk. AsLang-
landsandhiscolleagues[11]werefinishingtheirnumericalstudyofuniversalityandconformalinvariance
of crossing probabilities, I. Gelfand suggested to explore percolation on compact Riemann surfaces. The
simplestsurfacetostudyisthetorusandthemostnaturalobservableisthenthehomologicpropertiesof
thepercolatingcluster,ormoreprecisely,theprobabilitythataconfigurationcontainsahomologicallynon-
trivialcluster. (Sincetheseclustersaregeometricobjects,itmightbeeasiertothinkabouttheirhomotopic
propertiesinsteadoftheirhomologicalones.) Letω1andω2bethetwo-dimensionallinearlyindependent
vectorsalongthetwosidesoftheparallelogramdefiningthetorus. Inthefollowingthesewillbeidentified
topointsinthecomplexplane. Ifanon-trivialclusterexistsandifitwindsatimesalongω1 ofthetorus
whileitwrapsbtimesalong−ω2,theclusterissaidtobeoftype{a,b}.Allothernon-trivialclustersofthat
configuration,ifany,willbeofthesametype.(Theintegersaandbarecoprimes.Types{a,b}and{−a,−b}
areconsideredidentical.) Forthatreason,thehomologypropertyofaconfigurationmaybedefinedasthe
typeofitsnon-trivialclusters. Iftheconfigurationcontainsnonon-trivialcluster,itissaidtobeoftype{0}.
Finally,iftheconfigurationcontainsaclusterthathasbothapatharoundthefirstcycle,thatisalongω1,
andapathalongω2,thisconfigurationisoftypeZ×Z.Withthatnotation,eachconfigurationisassociated
withoneofthesubgroupsHofthehomologygroupZ×Zofthetorus: {0},Z×Zand{a,b}witha,bco-
primes. Thesamenotation{a,b}isusedforthetypeofaconfigurationandthesubgroupgeneratedbyan
elementofthattype. Langlandsetalmeasuredtheprobabilityofafewofthesesubgroupsforpercolation
andgavesomenumericalevidencefortheirconformalinvariance.
Pinson [17] obtained analytic expressions for the probability of these various subgroups as functions
ofthequotientτofthefundamentalperiodsω1,ω2 ∈ Cofthetorus. Hiscomputationreliesonaclever
argument giving an orientation to the curves bounding clusters. (See [14, 9].) This is done in a way that
doesnotchangethepartitionfunction,butdoesallowfortheidentificationofthehomologypropertiesof
interveningclusters. Hiscomputationismathematicallyrigorous,exceptforthesteptakingthelimitasthe
meshgoestozero;forthis,heusedNienhuis’renormalizationgroupargument[14]thattiesthequantities
understudytoknownresultsfortheCoulombgas. Amorerigoroustreatmentofthisstepremainsopen.
Arguin [1] extended Pinson’s argument to Q-Potts models, Q = 1,2,3,4. To do so, he considered the
Fortuin-Kasteleyngraphsorclustersofconfigurations. Thesearethenaturalextensionsoftheclustersof
percolation,thePottsmodelwithQ=1. ArguinshowedthatPinson’sformulaeneedonlyasmallchange
fortheQ-PottsmodelwithQ ≥ 2. Healsosupportedhisnewexpressionwithnumericaldataforthefour
integervaluesofQ.
WorksonorusingprobabilitiesofhomologysubgroupsofFKclustershasnotbeenlimitedtothethe-
oreticalpredictions. Ziff,Lorenz,Kleban[23]werethefirsttoprovidenumericalsupportfortheiruniver-
sality. Later Newman and Ziff [13] used them to give a precise estimate of the critical probability for site
percolation on a square lattice. It was then the most precise available estimate. And recently they were
2
againusedtoobtainpreciseestimatesforcriticalprobabilityforpercolationonseverallattices[7]. (These
probabilitiesarecalledwrappingprobabilitiesintheseworks.)
InthedefinitionofPottsmodels,Qgivesthenumberofstatesaccessibletothebasicrandomvariables,
often called spins. As such, Q must be an integer. When the partition function is rewritten in terms of
Fortuin-Kasteleyngraphs(hereafterFKgraphs),theparameterQappearsintheBoltzmannweightasQNc
whereNc isthenumberofFKconnectedcomponentsintheconfiguration. Inthisformulation,thecondi-
tionthatQbeanintegermayberelaxed. Onethengetsaone-parameterfamilyofmodels,usuallystudied
for the values of Q in the interval (0,4]. It is between this family of models and the family of stochastic
Loewnerprocessesthataclosetieseemstoexist,andhasbeenestablishedforsomeparticularcases. The
stochasticLoewnerequationwithparameterκ(SLEκ)isbelievedtodescribethegrowthoftheboundary
ofaFKgraph. TheexactrelationshipbetweenthetwoparametersQandκis
4π
Q=4cos2
κ
withκ ∈ [4,8)and,again,Q ∈ (0,4]. Percolationcorrespondstoκ = 6(andQ = 1)andtheIsingmodelto
κ = 16 (Q = 2). ThemathematicaltoolstodescribenotonlytheboundaryofasingleFKcluster, butthe
3
setofloopsdescribedbytheboundaryofallclustersinaconfigurationarenowemerging. Conformalloop
ensembles,definedbyCamiaandNewmanforpercolation[4]andmoregenerallybyWerner[22](seealso
[3]),mightallowfortherigorousstudyofhomologicalpropertiesofconfigurations,asdefinedandstudied
byLanglandsetal,PinsonandArguin.
Thegoalofthepresentpaperistoextractfromtheknownexpressionsoftheprobabilitiesforthevari-
oushomologysubgroupstheirasymptoticbehaviorfortwolimitingcases. Thefirstiswhenthequotientτ
oftheperiodsgoestoinfinityortoarealrationalnumber. ThesecondiswhenQgoestozero. Thereason
tostudythelatterismostlycuriosity. Fortheformer,thereasonistwofold. ManyresultsprovedusingSLE
techniquesdescribeasymptoticbehavior.Thefirstreasonisthereforetoseekexponentstodescribelimiting
behaviorthatmightbeeasiertoobtainwithSLE(orCLE).Thesecondreasonistoprobedeepertherela-
tionshipbetweenSLEandconformalfieldtheory(CFT).Severalcriticalexponentsappearing(rigorously)
inthecontextofSLEhadbeenpredictedwithinCFT,andalargesubsetoftheseappearedintheKactable
oftheassociatedminimalconformalmodel. Itisagreed,butnotproved,thatSLEκ describespropertiesof
theconformaltheorywithcentralcharge
(cid:18) (cid:19)
κ 4
c(κ)=13−6 + .
4 κ
Minimalmodelsappearwhencandκarerational. Letκberationalandoftheform4p(cid:48)/pwithp(cid:48) >p≥1,
coprimeintegers.Theconformalspectrumoftheminimalmodelwithcentralchargec=c(κ)isconstructed
fromtheVirasorohighestweights
(κr−4s)2−(κ−4)2
hr,s = 16κ , 1≤r≤p−1, 1≤s≤p(cid:48)−1. (1)
Ithasbeenrecognizedhoweverthattheminimalmodels,constructedoutoffinitesetsofprimaryfieldsand
thereforeofhighestweightshr,s,areprobablytoorestrictiveandmightnotcaptureallphysicalobservables.
Half-integersrandshavebeenconsidered[20]andseveralworksaboutlogarithmicminimalmodelshave
shownthattheupperboundsonrandsneedtoberelaxed.(See,forexample,[12,15]forrecentarguments.)
Maybe one of the most striking examples of this fact is Cardy’s formula that describes the probability of
crossing within a rectangle for percolation. For limiting geometries, that is for rectangles very wide or
narrow,theprobabilitiesapproach0or1withthepowerofh1,3 = 13,anexponentthatdoesnotbelongto
theminimalset. Anotherexampleisrelatedtotheproblemstudiedinthepresentnote. In[2],Arguinand
Saint-Aubinshowedthat,whenthequotientτofthefundamentalperiodsofthetorustendstozeroalong
theimaginaryaxis,theprobabilityπ({1,0})fortheIsingmodelgoesto1asintuitivelyitshould,butmore
preciselyitgoesasπ({1,0})→1−(q2)81f1(q2)−(q2)31f2(q2)−... whereq=eiπτandf1andf2areanalytic
inaneighborhoodofq = 0. Theexponentsaretwicethehighestweightsh1,2 = 116 andh3,3 = 16;thefirst
belongstothespectrumoftheminimalmodel,theseconddoesnot. Itisthisobservationthatledustoask
3
2
5
3
4
3
Τ Τ(cid:43)1
1
2
3
Ω
2
1
3
Ω
1
(cid:45)5 (cid:45)4 (cid:45)1 (cid:45)2 (cid:45)1 1 2 1 4 5
3 3 3 3(cid:45)1 3 3 3 3
3
Figure1: Thetorusinthecomplexplane,withτ=−2/3+i
whetherexponentsobtainedbytakinglimitsofthegeometrywouldalwaysbeintheextendedKactableof
thecorrespondingmodelswhenκisrational. (Everyconformalweighthr,s isrepeatedaninfinitenumber
oftimesintheextendedKactable. ArguinandSaint-Aubinchose(r,s) = (1,2)and(3,3)fortheleading
exponents of the Ising model. We shall come back to this choice after determining the exponents for the
generalcase.)
Ournotationsarethefollowing. ThetorusisidentifiedwiththequotientC/{ω1,ω2}where{ω1,ω2}is
the integral lattice generated by ω1,ω2 ∈ C such that 0,ω1,ω2 are not colinear. We choose ω1 = 1 and
Imω2 >0. Theirquotientτ=ω2/ω1isthemodulusofthetoruswithτrandτi >0itsrealandimaginary
parts. Figure 1 shows these basic elements for a torus with τ = −2 +i. We follow the convention set in
3
[17, 1] for the winding numbers: they are positive in the direction of ω1 and −ω2. Figure 2 shows FK
configurationsofthreedifferenttypesdrawnonthetorusτ = i. Configuration(c),forexample,isoftype
{2,−1}accordingtotheaboveconvention.
Itisnaturaltobreakthepartitionfunctionintosumsoverconfigurationsofagiventypeorgenerating
(a) (b) (c)
Figure2: ExamplesofFKconfigurationsoftype(a){0},(b)Z×Zand(c){2,−1}groups,drawnonthetorus
withτ=i
4
agivensubgroupH. Ifa∧bdenotesthegreatestcommondivisorofaandb(witha∧0=aforalla),the
partitionfunctionis (cid:88)
Z=Z({0})+Z(Z×Z)+ Z({a,b}). (2)
a∧b=1
The observables under study are the probability of a given subgroup H, namely π(H) = Z(H). All these
Z
quantitiesdependonthesizeofthelatticecoveringthetorusandthemodellabelledbyQ. (Forclaritywe
sometimesaddanindex,Qorτ,toquantitiesunderstudy,e.g.Z=ZQ.) Theirthermodynamiclimit,when
themeshsizegoestozero,areknownatthecriticaltemperature. TheexpressionsobtainedbyPinson[17]
forQ=1andgeneralizedbyArguin[1]forQ∈{1,2,3,4}are
(cid:88)
Z({a,b})= Zbk,ak(g/4)(cos[πe0k]−cos[πk]) (3)
k(cid:15)Z(cid:88)
Z({0})= 12 Zm,m(cid:48)(g/4)cos[π(m∧m(cid:48))] (4)
m,m(cid:48)(cid:15)Z
Z(Z×Z)=Q×Z({0}) (5)
where
(cid:114)
1 g
Zm,m(cid:48)(g)= |η(q)|2 τ e−πg|mτ−m(cid:48)|2/τi (6)
i
and
Q=4cos2[πe0/2], g=4−2e0, e0 =2−8/κ, Q∈(0,4], e0 ∈[0,1), κ∈[4,8). (7)
TheparametersQ,g,e0andκareallinone-to-onecorrespondencetooneanotherintheirrespectiverange.
(We u(cid:81)se them in the way historical developments have introduced them.) Dedekind function is η(q) =
q1/24 (1−qn). Pinson’sandArguin’sargumentsextendtriviallytothemodelsofFortuin-Kasteleyn
n∈Z
clusterwitharealQintheinterval(0,4]. Weusetheseexpressionsasourstartingpoint.
The paper is organized as follows. In the next three sections, we study the following three limits: of
π({1,0})whenτ = iτi andτi → ∞,ofπ({a,b})whenτ = dc +iτi withτi → 0andfinallyofπ(H)forany
H⊂Z×ZwhenQ→0. ThelastsectionisdevotedtoMonteCarloverificationsofsomeoftheresults.
2 The probability π({1,0}) in the limit τ = 0, τ → ∞
r i
The first limit to be studied is when τ = iτi with τi → ∞, i.e. the limit when the torus becomes a very
thin ring. The corresponding parallelogram in the complex plane becomes an infinitely tall rectangle of
constantwidthequalto1. Curveswindingoncealongω1 becomeverylikely. Infacttheirrelativelength
with respect to those winding once in the direction ω2 becomes negligible and it is therefore expected
that,inthislimit,allconfigurationswillhavecurvesoftype{1,0}andnoneoftype{0,1}. Inotherwords,
π({1,0}) → 1 and the probability of all other groups goes to 0. What should be the expected behavior of
π({1,0}) for finite but very large τi? Cardy’s formula [5] provides a fair guess. This formula gives, for
percolation,theprobabilityπh ofhorizontalcrossinginarectangleofwidthHandheightV asafunction
oftheaspectratior=V/H. Forlimitinggeometriestheprobabilitybehavesas
πh(r)−→c1e−π/3r and 1−πh(r) −→ c2e−πr/3
r→0 r→∞
for known constants c1 and c2. Even though the intersections of a percolating cluster with the left and
right edges of the rectangle might be in general at different height, these two intersections are likely to
havepointswiththesameverticalcoordinatesiftherectangleisverynarrow,thatiswhenr → 0. Sucha
percolatingclusterwouldbeaFKclusteroftype{1,0}, ifoppositeedgesoftherectanglewouldbeglued
together. Thereforeonemayexpectthefollowingbehavior
(cid:88)
π({1,0})=1− cnqγn (8)
n
5
withpositiveexponentsγnandthenaturalparameterq=e−2πτi iftherealpartofτvanishes. Notethatq
goesto0whenτi →∞. ThegoalofthissectionistodeterminetheleadingexponentsγnasafunctionofQ
or,equivalently,e0. (Somecareshouldbeexercisedastheimmediateextensionofqtoaτintheupper-half
plane by q = e2πiτ does not coincide with the usual definition of the nome of elliptic functions which is
eπiτ.)
The probability π({1,0}) is given in the form ZQ({1,0})/ZQ. The first step is to express the numerator
anddenominatorinaformsuitabletoextracttheseexponents. From(3):
(cid:88) (cid:114) (cid:88)
ZQ({1,0})= Z0k,1k(g/4)(cos[πe0k]−cos[πk])= |η(q1)|2 4gτ e−π4gτki2(cos[πe0k]−cos[πk]).
k(cid:15)Z i k(cid:15)Z
1
Torewritetheeτi intermsofq,Poissonsummationformulawillbenecessary:
(cid:88) (cid:88)
1
e−πan2+bn = √ e−πa(k+b/2πi)2. (9)
a
n(cid:15)Z k(cid:15)Z
Afterexpandingthecosinesintermsofexponentials,Poissonformulagives
(cid:88)
1
ZQ({1,0})= |η(q)|2 (q2(k+e0/2)2/g−q2(k+1/2)2/g). (10)
k(cid:15)Z
Since the function q−1/24η(q) has a Taylor expansion, the above form allows for the identification of the
leading terms in the numerator. Note however that the expansion of |η(q)|2 will not be used, since this
samefactorappearsinthedenominator.
Thedenominator (cid:88)
Z =(Q+1)Z({0})+ Z({a,b})
Q
a∧b=1
has two parts, which will be tackled separately. The partition function restricted to configurations with
onlytrivialclustersis
ZQ({0})= 2|η(1q)|2(cid:114)4gτ (cid:88) e−πg(m4ττ2ii+m(cid:48)) cos[π(m∧m(cid:48))].
i m,m(cid:48)∈Z
Togetridofthecos[π(m∧m(cid:48))],wenoticethat
(cid:88) (cid:88) (cid:88) (cid:88)
(cid:16) (cid:17)
= + − (11)
m,m(cid:48)∈Z m,m(cid:48)∈2Z m,m(cid:48)∈Z m,m(cid:48)∈2Z
In the first sum, both m and m(cid:48) are even which makes m∧m(cid:48) even and cos[π(m∧m(cid:48))] = 1. The other
terms,intheparenthesis,aretermsforwhicheithermorm(cid:48)isodd,andcos[π(m∧m(cid:48))]=−1. Therefore:
ZQ({0})= 4|η(1q)|2(cid:114)τg(cid:16)2 (cid:88) − (cid:88) (cid:17)e−πg(mτ42iτi+m(cid:48)2).
i m,m(cid:48)∈2Z m,m(cid:48)∈Z
Sumsovermultiplesofanintegerf∈Nwillappearoftenanditisusefultodefine
σ(f,g)=(cid:114)g (cid:88) e−πg(m24ττ2ii+m(cid:48)2)
τ
i m,m(cid:48)∈fZ
(cid:114) (cid:88) (cid:88)
= g( e−πgf42τmi (cid:48)2)( e−πgf24m2τi)
τ
i m(cid:48)∈Z m∈Z
(cid:88)
2 2m(cid:48)2+gf2m2
= q gf2 8 (12)
f
m,m(cid:48)∈Z
6
wherePoissonformula(9)wasusedagaininthelastline. ThepartitionfunctionZQ({0})isthen
(cid:88)
ZQ({0})= 4|η(1q)|2(2σ(2,g)−σ(1,g))= 2|η(1q)|2 (qm2g(cid:48)2+gm22 −q2mg(cid:48)2+gm82). (13)
m,m(cid:48)∈Z
TheremainingtermofZQ,thatincludesconfigurationswithnon-trivialFKclustersoftype{a,b}forall
aandbcoprimes,ismorecomplicated. Thesum
(cid:88) (cid:88)
ZQ({a,b})= Zm,m(cid:48)(g/4)(cos[πe0(m∧m(cid:48))]−cos[π(m∧m(cid:48))]) (14)
a∧b=1 m,m(cid:48)∈Z
contains two terms. The second with cos[π(m∧m(cid:48))] is exactly twice the partition function ZQ({0}) just
calculated. The first with cos[πe0(m ∧ m(cid:48))] does not simplify as easily; the sums must be reorganized
before(9)isused. Todoso,consider,formfixed,thefunctioncos[πe0(m∧m(cid:48))]. Whenmisnon-zero,itis
periodicinm(cid:48)withperiodm. Therefore
(cid:88) (cid:88) (cid:88)
Zm,m(cid:48)(g/4)cos[πe0(m∧m(cid:48))]= C(d,e0)Zm,m(cid:48)(g/4), m(cid:54)=0 (15)
m(cid:48)∈Z d|mm(cid:48)∈dZ
with (cid:88)
d
C(d,e0)= cos(d2πe0)µ(d ) (16)
d2|d 2
whereµ(x)istheMo¨biusfunctionofx. (Recallthatµ(1) = 1,µ(n) = 0ifnhasrepeatedprimefactorsand
µ(n)=(−1)(cid:96)ifnistheproductof(cid:96)distinctprimes.)Toget(15-16),thesumoverm(cid:48)wasdividedintosums
oversubsetswhichhavethesamevalueofcos[πe0(m∧m(cid:48))],inafashionsimilartothesplittingproposed
inequation(11). Thesesubsetsarecloselyrelatedtothedivisorsofm,thereforeleadingtothesplittinginto
sumsoverthemultiplesofthesedivisors. Wemuststress,however,thattheonlydivisorstobeconsidered
ind|marethepositiveones. Theremainingsumcanbewrittenwiththehelpof(15)as
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
Zm,m(cid:48)(g/4)cos[πe0(m∧m(cid:48))]= Z0,m(cid:48)(g/4)cos(πe0m(cid:48))+ C(d,e0)Zm,m(cid:48)(g/4)
m,m(cid:48)∈Z m(cid:48)∈Z m∈Z∗d|mm(cid:48)∈dZ
(cid:88) (cid:88) (cid:88) (cid:88)
= Z0,m(cid:48)(g/4)cos(πe0m(cid:48))+ C(d,e0)Zm,dm(cid:48)(g/4)
m(cid:48)∈Z m∈Z∗d|mm(cid:48)∈Z
where Z∗ = Z \ {0}. In the above expression, the terms with m = 0 get a special treatment because of
the particular definition of m∧m(cid:48) when m is 0. These were already encountered in the computation of
ZQ({1,0})andareequalto
(cid:88) (cid:88)
1
Z0,m(cid:48)(g/4)cos(πe0m(cid:48))= |η(q)|2 q2(k+e0/2)2/g.
m(cid:48)∈Z k(cid:15)Z
Forthetriplesum,thesumoverdivisorscanberearrangedusing
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
h(m,d)= h(m,d)= h(md,d)
m∈Z∗d|m d∈N∗m∈dZ∗ m∈Z∗d∈N∗
andsimilarlyforthesumofd2|dinC(d,e0).Thesemanipulationshavedoubledthenumberofsumsin(14)
fromtwo,onmandm(cid:48),tofour,onm,m(cid:48),d,d2. ThisisthepricetopaytousePoissonformulaonthesum
overm(cid:48)andcasteverythingintopowersofq. Theresultis
(cid:88)
Zm,m(cid:48)(g/4)cos[πe0(m∧m(cid:48))]
m∈Z∗,m(cid:48)∈Z (cid:88) (cid:88) (cid:88)
= C(d,e0)Zdm,dm(cid:48)(g/4) (17)
m∈Z∗d∈N∗m(cid:48)∈Z
= |η(q1)|2 (cid:88) (cid:88) (cid:88) cos(πed0dd2)µ(d)qg(md8d2)2+g(2dmd(cid:48)22)2 (18)
m∈Z∗d,d2∈N∗m(cid:48)∈Z 2
7
andthecompletepartitionfunctionZQis
(cid:88) (cid:88)
|η(q)|2ZQ = q2(k+e0/2)2/g+ (Q2−1) (qm2g(cid:48)2+gm22 −q2mg(cid:48)2+gm82)
k∈Z m,m(cid:48)∈Z
+ (cid:88) cos(πe0d2)µ(d)qg(md8d2)2+g(2dmd(cid:48)22)2. (19)
dd
m∈Z∗ 2
d,d2∈N∗
m(cid:48)∈Z
Theprobabilityπ({1,0})isthequotientofZQ({1,0})givenin(10)andofZQ.
Itisnowstraightforwardtoseethatthelowest-orderterminqis 2eg20 forboththedenominatorZQ and
the numerator ZQ({1,0}). After simplification of the common factor qe20/2g, an expansion can be done to
obtainthewholesetsofexponents. Anexhaustivelistofpossibleexponentsisgivenbytakingexponents
inthenumeratorandinthedenominator,plusanyintegrallinearcombinationsofthemwhicharisefrom
higherordertermsintheexpansion. Thepossibilitythatsomeofthemcouldhavevanishingcoefficientsis
notexcluded.
Itisinterestingtocomparetheleadingexponentswithvalues(1)givenbyCFTintheKactable[8]. In
termsofgande0theyare
[r−(g/4)s]2−e2/4
hr,s = g 0 (20)
forr,spositiveintegers. Notethat e20 ishalfthepowerofqthatwassubstractedtosimplifythenumerator
4g
anddenominator. Thefirstexponentsforπ({1,0})aregivenby
1−e2 1−e
γ1 = 4(2−e0), γ2 = 2−e0 (21)
0 0
and their integer multiples. On the range of e0, γ2 > γ1. The two exponents become equal in the limit
e0 =1(Q=0);thisparticularcasewillbestudiedinsection4.
Coincidences of these leading exponents or higher ones with elements from the Kac table, if any, will
occurintheformγ=2hr,sforsomer,sbecauseofthecontributionofholomorphicandanti-holomorphic
sectors. Such coincidences do occur. The simplest r and s giving γ1 are r = 21,s = 0 and, those giving
γ2, r = 0,s = 1. It is somewhat unusual to choose vanishing s or r. Recall however that, for logarithmic
minimalmodels,theKactableisextendedandtheperiodicityofelementshr,s = hr+p,s+p(cid:48) forthemodel
withκ = 4p(cid:48)/pallowstochooserandspositive. Forsomeminimalmodels, itishoweverimpossibleto
accountforγ1 withintegersrands. Half-integersmustbeused. ArguinandSaint-Aubin[2]identifiedthe
two leading exponents for the Ising model to 2h1,2 and 2h3,3. Note that, when either r or s is zero, then
hr,s = h−r,−s. Moreover, if half-integer indices are included, the periodicity property can be refined to
hr,s =hr+p/2,s+p(cid:48)/2. TheIsingmodelcorrespondstop=3,p(cid:48) =4andtheirexponentsarerelatedtoours
byh1,2 =h−1/2,0 =h1/2,0andh3,3 =h0,−1 =h0,1.
These two exponents γ1 and γ2 are related to the fractal dimensions of geometric objects, namely the
massandthehullofaclusterrespectively. (See[21,10,20]. Foranextensionofthesegeometricobjectsto
loopgasmodels,see[19].) IntheFKformulationoftheQ-Pottsmodels,theFKclustermassattachedtoa
siteisthenumberofbondsinthecomponentoftheFKgraphcontainingthissite. Intheplane,thehullofa
FKclusteristhesetofbondsthatcanbereachedfrominfinitywithoutcrossinganybondfromthecluster.
(Onatorus,eachclusterhasaninnerandanouterhull.) Theirfractaldimensionis2−2∆where∆ish
1/2,0
fortheclustermassandh0,1forthehull.
A natural explanation for h in the present context is provided by Cardy [6] (see also [20]). Note
1/2,0
firstthattheonlywaytokeepaconfigurationfromhavingaclusteroftype{1,0}istohaveaclusterinthe
verticaldirection. Itislikelythatitstypewillbe{m,1}forsomem∈ZorZ×Z. Cardygivesanexpression
fortheprobabilityP(n,k)ofhavingnclustersconnectingthetwoextremitiesofacylinderwhoselengthis
ktimestheperimeterofitssection. HefindslogP(n,k) ∼ −2π(n2− 1)kifn ≥ 2. Hepointsoutthatthis
3 4
expressionevaluatedatn = 1isnottheprobabilityofhavingasingleclusterbetweenthetwoextremities,
8
butitistheprobabilityofhavingasingleclusterbetweentheextremitiesthatdoesnotwindintheother
direction. Whenallconfigurationswithasingleclusterareconsidered,disregardingtheirbehaviorinthe
otherdirection,theprobabilityislargerandgivenbylogP(1,k) ∼ −52π4k. Becausee0 = 23 forpercolation,
ourfirstcorrectiontermisqγ1 =e−5πτi/24,inagreementwithhisresult.
Inarecentstudyofpercolation,Ridout[18]hasarguedthattheprimaryfieldresponsibleforchanging
boundaryconditionsinthecomputationofWatts’formulashouldbeφ . Thisidentificationforceshim
2,5/2
toshift,intheextendedKactable,theadmissiblevaluesofsby 1 whenriseven. Onewouldliketoseea
2
relationshipwithouridentificationofγ1 as2h1/2,0. Howeveritisrthattakesanhalf-integervalueinour
case,andsinhiscase. Moreoverthevalueh = 5 doesnotappearinhisshiftedextendedKactable.
1/2,0 96
Theotherexponentsinthenumeratorofπ({1,0})arealsopartoftheextendedKactable. Theyappear
withr = k+1,s = 1for 2(k+e0/2)2−e20/4 andr = k+1/2,s = 0for 2(k+1/2)2−e20/4. Notalltheexponents
g g
ofthedenominatorhoweverappearintheextendedKactable,evenifoneallowshalf-integersrors. For
examplethedenominatoroftheexponents g(2dmd(cid:48)22)2 appearinginthelastsumofZQ isnotbounded. There
is no hope to find them all in the extended Kac table. Could these terms drop out of the sum because of
cancellations? Thegeneralcaseisdifficulttoassess,butthishappensinsimplecases.
ItisindeedpossibletofindsimplerformforthedenominatorforthefourintegralvaluesQ = 1,2,3,4
thatcorrespondtoe0 = 32,12,13,0. Ford > 0andtheseparticularvaluesofe0 andfore0 = 1,thefunction
C(d,e0)isparticularlysimple:
C(d,0)=δd,1 (22)
1 δ 3δ
C(d,3)= d2,1 −δd,2− 2d,3 +3δd,6 (23)
1
C(d,2)=2δd,4−δd,2 (24)
2 3δ δ
C(d, )= d,3 − d,1 (25)
3 2 2
C(d,1)=2δd,2−δd,1. (26)
(cid:80)
TheproofoftheseformulaeisgivenintheAppendix.Thecontributionof m,m(cid:48)∈ZZm,m(cid:48)(g/4)cos[πe0(m∧
m(cid:48))]toZQisthenmuchsimpler. Itis
(cid:88) (cid:88)
1 (cid:16)qm12(cid:48)2+3m2 −q3m4(cid:48)2+m32(cid:17) forQ=1, 1 (cid:16)qm24(cid:48)2+6m2 −qm6(cid:48)2+3m22(cid:17) forQ=2,
2 2
(cid:88)
1 (cid:16)qm60(cid:48)2+15m2 −qm15(cid:48)2+154m2 −q3m20(cid:48)2+5m32 +q3m5(cid:48)2+51m22(cid:17) forQ=3
2
(cid:88)
and qm(cid:48)22+m2 forQ=4.
All the sums above are on m,m(cid:48) ∈ Z. It is then straigthforward to show that, upon simplification of the
(cid:80)
factor qe20/2g, these forms (and therefore ZQ) can be written as a product i,j(q2hifi(q2))(q2hjfj(q2))
of two finite sums where fi,fj are analytic in a neighborhood of 0 and where all the hi,hj belong to the
correspondingextendedKactableforsomerandsintegersintherange[0,p].
3 Probabilities π({a,b}) in the limit τ = c/d, τ → 0
r i
Theprobabilitiesπ({a,b})=πτ({a,b})alsohavealimit,either0or1,whenτapproachesarationalnumber
on the real line. To see this first intuitively, consider the case {a,b} = {−2,1}. Figure 3 presents two tori
whosemodulusparameterisontheline−2+iτi. Foreach,twoneighboringfundamentalparallelograms
havebeendrawn. Acurvelinkingtheorigintothevertexatz=2iτi,likethoseshown,isoftype{−2,1}. A
curveoftype{−2,1}doesnotneedtostartatavertex,ofcourse,butthosedrawnshowhowthecurvesof
thistypewillbeprevailing.Indeed,asthemodulusparameterτslidesdowntheverticallineτr =−2,these
9
2
1
Τ(cid:61)(cid:45)2(cid:43)iΕ,Ε(cid:174)0
(cid:45)2 (cid:45)1 1 2
Figure3: Curvesoftype{−2,1}becomemorelikelyasτapproachesτ∼−2+i0+
curves become very short and likely. Therefore the probability πτ({−2,1}) should converge to a number
largerthan0whenτ→−2. Thissectionshowsthatitactuallygoesto1.
We have identified a torus with its modulus τ, a complex number in the upper-half plane H. As it is
well-known, this correspondence is not unique, since any pair ω1(cid:48) = mω1 +nω2 and ω2(cid:48) = pω1 +qω2
withm,n,p,q ∈ Zandmq−np = 1describesthesametorus,butwithanewmodulusτ(cid:48) = ω(cid:48)/ω(cid:48). The
2 1
speciallineartransformations(q p)withintegercoefficientsanddeterminant1formthemodulargroup
nm
SL(2,Z). Itisgeneratedbytwomatrices
(cid:18) (cid:19) (cid:18) (cid:19)
0 −1 1 1
s= and t=
1 0 0 1
whoseactiononτis
τ(cid:55)→s −1/τ and τ(cid:55)→t τ+1. (27)
The probabilities π({0}) = πτ({0}) and π(Z×Z) = πτ(Z×Z) are invariant under the change of τ by an
elementofSL(2,Z),buttheprobabilitiesπτ({a,b})arenot. Arguin[1]gavetheirtransformationlaws
πτ({a,b})=πτ+1({a+b,b})=π−1/τ({−b,a}) (28)
or,equivalently
πτ({a,b})=πgτ(g·{a,b}), g∈SL(2,Z) (29)
whereτ(cid:55)→gτdenotestheactiondefinedby(27)andg·{a,b}standsforthematrixmultiplicationg(a).
b
Thesetransformationsfollowimmediatelyfromtheform(3)ofthepartitionfunctionZQ({a,b}).
Asimpleapplicationofthemodulartransformationgivesπ({0,1})intermsofπ({1,0}),namelyπτ({0,1})=
π ({1,0}). Theresultoftheprevioussectionimplieseasilythat
−1/τ
πτ=0+iτi({0,1})=πτ=0+i/τi({1,0})= ZQ(Z{1,0})(cid:12)(cid:12)(cid:12)(cid:12)
Q q(cid:48)
wherethepartitionfunctionsareevaluatedatq(cid:48) = e−2π/τi. Thelimitingbehaviorτi → 0+ willtherefore
becharacterizedbythesameexponentsobtainedforπ({1,0})whenτi →∞.
Letg∈SL(2,Z)andz(cid:55)→gztheassociatedmap. Itisconformal,one-to-oneonHandmapstherealline
ontoitself. Theimageundersuchamapoftheimaginaryaxiswillthereforebeacircleintersectingthereal
axisatrightangles.
Let{a,b}beapairofcoprimeintegers. Thenthereareintegerspandqsuchthatpa+qb=1. Therefore
g = (cid:0)ab−pq(cid:1) ∈ SL(2,Z). TheactionofgonHmapsapointτ = iτi,τi > 0,onthepositiveimaginaryaxis
intothepoint
a(cid:18)1−pq/abτ2(cid:19) i (cid:18) 1 (cid:19)
iτi (cid:55)→ b 1+p2/b2τ2i + τ b2 1+p2/b2τ2 . (30)
i i i
10