Orthogonal polynomial method and odd vertices in matrix mode

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