Orthogonal polynomial method and odd vertices in matrix models

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

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aORTHOGONALPOLYNOMIALMETHODANDODDVERTICESINMATRIXMODELSEttoreMinguzzi1DipartimentodiFisicadell’Universit`a,Pisa56100,ItalyandINFN,SezionediPisaAbstract.Weshowhowtousethemethodoforthogonalpoly-nomialsforintegrating,intheplanarapproximation,thepartitionfunctionofone-matrixmodelswithapotentialwithevenoroddvertices,oranycombinationofthem.1.IntroductionThemethodoforthogonalpolynomialsisapowerfultechniqueforthenonperturbativeintegrationofmatrixmodelsoverone[1]ormorematrices[2]inparticularwithevenpotential,i.e.withverticeswithanevennumberoflegs.Indeed,withevenpotential,thecalculationsim-pli esbothbecausetheintegralsarewellde nedand,asweshallsee,thenumberofequationsneededtosolvetheproblemissmaller.Ontheotherhandthemodelwithoddvertices,inparticularwithcubicverticesismorenaturalinanumberofproblems;e.g.inthedynamicaltriangulationmodelofquantumgravity,wheretherandomsurfaceisgivenbyapolyhedronwithtriangularfaces,theorderoftheverticesappearinginthedualgraphsisalwaysthree.Br´ezinetal.[3]solvedtheproblemwithcubicverticesusingthesaddlepointtechnique.Bessis[4]introducedanalternativemethod(theorthogonalpolynomialmethod)

whichtosomeextentappearsmorepowerfule.g.indealingwithma-trixmodelwithmorethanonematrixvariable[2].Inparticular,theorthogonalpolynomialmethodhasbeenprovedusefulinthetreatmentofacubicvertextwo-matrixmodel[5]inthecontextoftheIsingmodelonarandomplanarlattice.

Thepurposeofthispaperistoshow,inasystematicway,howtoextendtheorthogonalpolynomialmethodtoarbitraryvertices,bothevenandoddandanycombinationofthem.WeshallfollowthearticleofBessisetal.[1]generalizingsomeaspectstothecaseofoddvertices,inparticularweshallrecover,forthesimplestcaseofcubicvertices,theresultof[3]forsphericaltopology.Hopefullysuchatreatmentcanbeextendedtohighergenus.

Theuseofmixedverticese.g.cubicplusquarticvertex,allowsustowriteawellde nedi.e.convergent,partitionfunctionbyaddingtothecubicinteractionaquartictermwhichmakestheactionboundedfrom

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

2ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES

below,andthustheintegralgivingthepartitionfunctionwellde ned.Attheendonecantakethelimitwhenthecouplingconstantofthequarticvertexgoestozero.

Westartfromthepartitionfunction ZN(g)=dMe trS(M)(1)wheretheintegrationisoveranhermitianmatrixoforderNandwheretheactionisgiveningeneralby

1 S(M)=

jNj

Ni

ZN(0)=∞ h=0N2 2heh(g)(4)

where2 2histheEulercharacteristicoftheorientedribbongraphstobesummedintheperturbativeexpansionofthefunctionseh(g).Indeed,denotingsuchgraphswithcapitalletters,eachfunctionehadmitsthefollowingexpansion[1]

vi(G) igie

h

(g)=

Gconnectedofgenush

Anautomorphismofanorientedribbongraphisde nedinthefollowingway.Firstletusidentifytheorientedribbongraphasacommongraphplusacyclicorderingonthesetsofhalf-edgesattachedtoeachvertexandthende neanau-tomorphismoftheorientedribbongraphasanautomorphismofthegraphwhichleavestheorderingofeachvertexunchanged.It’sclearthattheautomorphismmustsendeachvertexintoavertexofequalvalence.2

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES3Figure1.Secondandfourthorderconnectedgraphs

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

4ORTHOGONALPOLYNOMIALMETHODANDODD

VERTICES

Figure2.Firstandsecondorderconnectedgraphs

withquarticvertex.

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES5

2.Themethodoforthogonalpolynomials

Achangeofintegrationvariablesin(1)leadsustotheintegrationovertheeigenvaluesλiofthediagonalmatrixλ trS(M)2 iS(λi)ZN(g)=dMe=kHdλi (λ)e(6)where (λ)=α<β(λβ λα)istheVandermondedeterminant.WeobtainthevalueoftheconstantkHusingtheresultsin[6]:kH=

.AsweseetheargumentoftheintegralistheproductoftheVandermondedeterminantsquaredandafactorizablefunctionoftheeigenvalues,thisfeaturemakestheorthogonalpolynomialmethodapplicable.Letusintroducethemeasuredµ(λ)=dλe S(λ),andtheorthogonalpolynomialsPn(λ)

+∞

dµ(λ)Pn(λ)Pm(λ)=hnδnm(7)j=1π N2 N iNj!

∞

wherePn(λ)isnormalizedbytheconditionthatthecoe cientofthetermwithhighestdegreeequals1

Pn(λ)=λn+....(8)

ThepolynomialsPn(λ)canbeobtainedinaconstructingwaye.g.bytheGram-Schmidtorthogonalizationprocedurefromthemonomials1,λ,λ2,....Asimpleanalysisofthisprocedureshowsthatthepolyno-mialsPjhavethewellde nedparity( 1)jiftheactionS(λ)iseven.Everypolynomialofdegreencanberewrittenasalinearcombina-tionofPmwithm≤n.TheVandermondedeterminantin(6)canberewrittenas

j 1 =det λi =det Pj 1(λi) = σ( 1)p(σ)N iPσ(i) 1(λi)(9)

wherethesecondequalityisduetothefactthataddingtoacolumnalinearcombinationoftheothercolumnsdoesnotchangethedetermi-nantofthematrix;( 1)p(σ)standsforthesignofthepermutationσ.Wecantakeadvantageofthecouplingoftheorthogonalpolynomialsdue 2in(6)toobtainthepartitionfunctionintermsofthenormoftheorthogonalpolynomials

N dµ(λi)Pσ1(i) 1(λi)Pσ2(i) 1(λi)( 1)p(σ1)( 1)p(σ2)ZN(g)=kH

σ1,σ2

=kH

σ1,σ2 i( 1)p(σ1)( 1)p(σ2)δσ1σ2 ihσ1(i) 1=kHN!h0h1...hN 1.

(10)

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

6ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES

Letusrewritethisexpressioninadi erentform.Thefollowingequa-tionisvalid

λPn(λ)=Pn+1(λ)+AnP(λ)+RnPn 1(λ)(11)

wherethetermswithindexlessthann 1areabsentbecauseaftermoltiplicationbyλtheydonotreachthedegreenandthusareorthog-onaltoλPn.ForparityreasonswhenS(λ)isevenAnvanishes.WeshallrefertotheprecedingequationasthestepequationbecauseitsrepeatedapplicationenablesustocalculateλiPn(λ)usingananalogywithallpossiblestaircasesistepslong.Thismethodwillbedevelopedinthefollowing

hn+1= section.Sincedµ(λ)Pn+1λPn(λ)(12)

= dµ(λ)[Pn+2(λ)+An+1Pn+1(λ)+Rn+1Pn(λ)]Pn(λ)=Rn+1hnthepartitionfunctioncanberewrittenas

ZN(g)=kHN!hN0RN1 1...R2

N 2RN 1(13)

whereh0=dλe S(λ).BeforepassingtothelimitforlargeN,wemustcompute

E1

N(g)=Z=1Rn(g)

N(0)N lnNlnh0(g)

(G) FgViii

h0(0)=

Gconnected Nχ

2 1)≤ 1,wehave

1

(0)=O(N 2

h)(16)

whichvanishesforlargeN.Thus

e0(g)=1

Nlim→∞EN(g)=Nlim→∞N lnRn(g)

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES7

3.Thenumberofstaircases

iWeshallneed,inthefollowing,thequantitiesβnde nedby

ihnβn=dµ(λ)Pn(λ)λiPn 1(λ).(18)

Wedevotethepresentsectiontothecalculationoftheaboveintegral.TocomputeλiPn 1wetakeadvantageofananalogywithallstaircasesofisteps;whereeachstepcangoup,comedown,orstayatthesamelevel.Theanalogycomesfromarepeatedapplicationofthestepequation.Aftertheintegrationonlythestaircaseswhichendonestepup,contribute.Eachofthemrepresentsaproductoffactors:ifastepisdownfromlevelntotheleveln 1weadda

factorRn,andifitstaysatthesamelevelnweaddafactorAn.Figure3showsanexampleofthiskindofcalculation.Sinceeverycoe cientAj,Rj,isafunction

iFigure3.βncomputedfromthestaircases.

ioftheindexjitwouldbedi culttohandthe nalexpressionforβn;

luckily,asweshallsee,theplanarlimit(N→∞)willenableustoneglectthedi erencesamongthesequantitiesrelativetodi erentilevels.InthislimitwemustcomputetheexpressionforβnsupposingthateachstepdownyieldsafactorR,andeachstepthatstaysatthesamelevelyieldsafactorA.Thusthequestionis:Howmanyarethestaircasesofistepswhose nale ectistogouponestep?LetjbethestepsoftypeA,thentheotheri jaredividedinpstepsupandp 1steps downsothati=j+2p 1.WithouttheAstepsthere2p 1arepstaircasesof2p 1stepswhose nale ectistogouponestep.InsidethesestaircaseswewanttoinserttheremainingjlevelsoftypeA:thereare2p places wheretheycanbeinserted,and,fora xed2p+j 1staircase,therearechoices.Finallythenumberofstaircasesj

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

8ORTHOGONALPOLYNOMIALMETHODANDODDVERTICESofistepswhose nale ectistogouponestepis

[i+1

(i 2p+1)!p!(p 1)!,(19)

ithecontinuumwhere[]standsfortheintegerpart,and,denotingbyβivalueofβn,wehave

[i+1

i=β(i 2p+1)!p!(p 1)!Ai 2p+1Rp 1(20)

wherethetilderemindsthereplacementAj→A,Rj→R.Thevalues iforthe rstfewiareofβ

2=2Aβ

3=3A2+3Rβ

4=4A3+12AR.β

Analogouslywede ne

ihn+1γn=(21) dµ(λ)Pn+1(λ)λiPn 1(λ).(22)

iBythesametechniqueusedforβnwe nd

[i

γ i=

Finallywede ne(i 2p+2)!p!(p 2)!Ai 2p+2Rp 2.(23)

[i+1

i=β i Rγ i 1=δ i 1 Aβ(i 2p+1)!(p 1)!2Ai 2p+1Rp 1.

(24)

iforthe rstfewiareThevaluesofδ

3=A2+2Rδ

4=A3+6ARδ

5=A4+12A2R+6R2.δ

4.Derivationofthecontinuumequations

InthissectionweshallexaminethecontinuumlimitN→∞,whichwillallowustowriteasimpleexpressionforthegeneratingfunction(25)

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES9e2(g)oftheplanargraphs.ThiswillalsojustifythereplacementAj→A,Rj→Rusedintheprevioussection.Letusconsidertheidentity ′nhn=dµ(λ)λPn(λ)Pn(λ)

′=dµ(λ)Pn(λ)[Pn+1(λ)+RnPn 1(λ)+AnPn(λ)]

′=Rndλe S(λ)Pn(λ)Pn 1(λ)(26) =Rndλe S(λ)S′(λ)Pn(λ)Pn 1(λ)

=(1 k i=3i 1g¯iβn)hnRn,

whereinthelastbutoneequalitywehaveintegratedbypartsandiinthelastequalitywehaveusedthede nitionofβn.Thuswehaveobtainedthe rstrecursionrelation

n=(1 k i=3i 1g¯iβn)Rn.(27)

Fromthisequationweinferinparticularthat:Rn(0)=n.Wewantto ndasecondrecursionrelationwhichrelatesthecoe cientsAnandRn.Weobservethat:

′dλe S(λ)λPn(λ)Pndλe S(λ)Pn(λ)λS′(λ)Pn+1(λ)+1(λ)=

=(An+An+1 k i=3(28)ig¯iβn+1)hn+1.

But ′′dλe S(λ)λPn(λ)Pndλe S(λ)Pn 1(λ)Pn+1(λ)=nAnhn+Rn+1(λ)

=nAnhn+Rndλe S(λ)Pn 1(λ)S′(λ)Pn+1(λ)

=nAnhn hn+1Rn

Asaresult,thesecondrecursionrelationis

(An+An+1 k i=3ig¯iβn+1)Rn+1=nAn Rn+1Rnk i=3i 1g¯iγn.k i=3i 1.g¯iγn(29)(30)

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

10ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES

Now,weextracttheplanarcasetakingthelimitN→∞.Letusintroducethesubstitutionsn

NA →R(x)(32)n

N →A(x)(33)

toobtain,takingintoaccountthepowerofNcontaineding¯i,thetwocontinuumequations

x=R(x) 1 k

giβ i 1(x) (34)

i=3

A(x k

)=gi δi(x),(35)

i=3

whereβ i(x)andδ i(x)areexpressedintermsofA(x)andR(x)asgivenbyeqs.(20,24).One nds,fromeqs.(27,30),thatthecontinuoussolutionA(x),R(x),isrelatedtothecoe cientsAnandRnby

Rn

A +On N 1

N=A Nn

n(g)

N 1R,(37)

n(0)=R

andthefunctione0(g)can

e(g)= berewritten,inthelimitN→1

dx(1 x)ln ∞,asR(x)

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES11su ces.Thequarticcasecanbeexplicitlyintegrated[1]toobtain

∞ 1(2k 1)!e

(g4)=(3g4)k(a 1)(a 9)=24k=1

1 12g4

(k+1)!(k 1)!.(41)

Recallingtheformulaforthetopologicalexpansion(5),onehastheinterestingequation

1(42)k!(k+2)!Gplanar,connected,withkquarticvertices

thatcanbecheckedfork=1andk=2usingthecontentsof gure2.Theradiusofconvergenceis1/12andg4c=1/12isthecriticalpoint.Forg4→g4coneobtainsthecriticalbehavior

e0(g4)～(g4c g4)5

=1 2g3AR

2g3R=A g3A2.

22g3x+σ(1+σ)(1+2σ)=0.(44)Infact,letusintroducethenewvariableσ= g3Arelatedtoxby(45)

2Thefunctionσ(x)=σ¯(g3x)isthesolutionof(45)whichvanishesfor

x=0;indeedwheng3=0thepotentialhasnolongeroddverticesandthenA(x)=0 σ¯(0)=0.Ourfunctione0(g3)mayberewritten,goingoverfromthevariablextothevariableσandintegratingbyparts

e0(g3)= 1

dx(1 x)ln

10 R(x)1+2σ(46)

= 3(1+σ1)(1+2σ1)2

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

12ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES

2whereσ1=σ¯(g3)isthesolutionof

22g3+σ(1+σ)(1+2σ)=0,(47)

whichvanishesing3=0.σ1canbeexpressedasanexpansioninpowersofg3usingLagrangetheorem,obtaining (3k 1)12 .(48)σ1= (k+1)2

Exceptforsomefactors,due

todi erentde nitions,ourresultscoincidewiththoseofBr´ezinetal.[3].Thepowerexpansionseriesfortheplanargeneratingfunctionis

e0(g3)=1

Γ(k+3)Γ(k/2+1)(49)

and,recallingthetopologicalexpansionfore0(g3),wereachtheformula

1Γ(3k/2)

Gplanar,connected,with2kcubicvertices2

Γ(3k/2)

2

√√3)kk 127√

212

weconcludethatthecriticalexponentremainsunchangedfromthequarticcase.

In2D-Gravity,wherethecontinuumsurfacesarereplacedbypoligo-nalizations,sucharesultisacheckoftheindependenceofthepartitionfunction,inthelimitofin nitenumberofvertices,ofthekindofpolig-onalizationonechoosestoapproximatethecontinuumsurfaces[7].

6.Conclusions

Indealingwithmatrixmodelsusuallyoneencountersmatrixmodelswithevenpotentialsothequestionnaturallyarisesifthereissomeobstructiontotheoddvertexcase.Inthispaperwehaveshownthat,evenif,intheoddvertexcase,theoriginalpartitionfunctionisillde ned,themethodoforthogonalpolynomialscanbeoftenappliedinitsmostnaiveform,thatisignoringallconvergenceproblems.Thisisjusti edbyaddingaregulatingevenvertextotheoddone,andtakingeventuallythelimitforitscouplingconstantgoingtozero.Wehaveextendedtheorthogonalpolynomialmethodtoanycombinationofoddandevenvertices,writingthetwoneededcontinuumequation.Theexplicitapplicationtothecubicvertexcasehasbeengiven,recovering

We show how to use the method of orthogonal polynomials for integrating, in the planar approximation, the partition function of one-matrix models with a potential with even or odd vertices, or any combination of them.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES13theresultofBr´ezinetal.[3].Anexplicitintegrationof3+4or5vertexcaseappearsfeasiblealongtheselinesandwouldbeausefulcheckoftheuniversalityofthecriticalbehavior.

Thegeneralsettingexplainedherecanbereadilydeveloped,intheplanarcase,alsofortwo-matrixmodelswithcouplingintheformoftheItzykson-Zuberformula[8,2],thecubiccasebeingalreadysolvedin[5].Furtherextensioncanbedevelopedinhighergenuscasese.g.inthecubiccaseforthetorustopology.

Acknowledgements

IamgratefultoP.Menottiforsuggestingthisproblemandforusefuldiscussions.

References

[1]D.Bessis,C.Itzykson,J.B.Zuber,Adv.Appl.Math.1(1980),109

[2]M.L.Metha,Commun.Math.Phys.79(1981),327

[3]E.Br´ezin,C.Itzykson,G.Parisi,J.B.Zuber,Commun.Math.Phys.59(1978),

35

[4]D.Bessis,Commun.Math.Phys.69(1979),147

[5]D.V.Boulatov,V.A.Kazakov,Phys.Lett.B126(1986),379

[6]M.L.Metha,RandomMatrices,NewYork,AcademicPress1967

[7]P.Ginsparg,G.Moore,Lectureson2DGravityand2DStringTheory,preprint

hep-th/9304011(1993)

[8]C.Itzykson,J.B.Zuber,J.Math.Phy.21(1980),411

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