Orthogonal polynomial method and odd vertices in matrix mode(12)

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

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35

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

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

hep-th/9304011(1993)

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