Orthogonal polynomial method and odd vertices in matrix mode(11)

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