Orthogonal polynomial method and odd vertices in matrix mode(5)

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)