Orthogonal polynomial method and odd vertices in matrix mode(9)

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)