Orthogonal polynomial method and odd vertices in matrix mode(4)

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.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES5

2.Themethodoforthogonalpolynomials

Achangeofintegrationvariablesin(1)leadsustotheintegrationovertheeigenvaluesλiofthediagonalmatrixλ trS(M)2 iS(λi)ZN(g)=dMe=kHdλi (λ)e(6)where (λ)=α<β(λβ λα)istheVandermondedeterminant.WeobtainthevalueoftheconstantkHusingtheresultsin[6]:kH=

.AsweseetheargumentoftheintegralistheproductoftheVandermondedeterminantsquaredandafactorizablefunctionoftheeigenvalues,thisfeaturemakestheorthogonalpolynomialmethodapplicable.Letusintroducethemeasuredµ(λ)=dλe S(λ),andtheorthogonalpolynomialsPn(λ)

+∞

dµ(λ)Pn(λ)Pm(λ)=hnδnm(7)j=1π N2 N iNj!

∞

wherePn(λ)isnormalizedbytheconditionthatthecoe cientofthetermwithhighestdegreeequals1

Pn(λ)=λn+....(8)

ThepolynomialsPn(λ)canbeobtainedinaconstructingwaye.g.bytheGram-Schmidtorthogonalizationprocedurefromthemonomials1,λ,λ2,....Asimpleanalysisofthisprocedureshowsthatthepolyno-mialsPjhavethewellde nedparity( 1)jiftheactionS(λ)iseven.Everypolynomialofdegreencanberewrittenasalinearcombina-tionofPmwithm≤n.TheVandermondedeterminantin(6)canberewrittenas

j 1 =det λi =det Pj 1(λi) = σ( 1)p(σ)N iPσ(i) 1(λi)(9)

wherethesecondequalityisduetothefactthataddingtoacolumnalinearcombinationoftheothercolumnsdoesnotchangethedetermi-nantofthematrix;( 1)p(σ)standsforthesignofthepermutationσ.Wecantakeadvantageofthecouplingoftheorthogonalpolynomialsdue 2in(6)toobtainthepartitionfunctionintermsofthenormoftheorthogonalpolynomials

N dµ(λi)Pσ1(i) 1(λi)Pσ2(i) 1(λi)( 1)p(σ1)( 1)p(σ2)ZN(g)=kH

σ1,σ2

=kH

σ1,σ2 i( 1)p(σ1)( 1)p(σ2)δσ1σ2 ihσ1(i) 1=kHN!h0h1...hN 1.

(10)