Orthogonal polynomial method and odd vertices in matrix mode(6)

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.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES7

3.Thenumberofstaircases

iWeshallneed,inthefollowing,thequantitiesβnde nedby

ihnβn=dµ(λ)Pn(λ)λiPn 1(λ).(18)

Wedevotethepresentsectiontothecalculationoftheaboveintegral.TocomputeλiPn 1wetakeadvantageofananalogywithallstaircasesofisteps;whereeachstepcangoup,comedown,orstayatthesamelevel.Theanalogycomesfromarepeatedapplicationofthestepequation.Aftertheintegrationonlythestaircaseswhichendonestepup,contribute.Eachofthemrepresentsaproductoffactors:ifastepisdownfromlevelntotheleveln 1weadda

factorRn,andifitstaysatthesamelevelnweaddafactorAn.Figure3showsanexampleofthiskindofcalculation.Sinceeverycoe cientAj,Rj,isafunction

iFigure3.βncomputedfromthestaircases.

ioftheindexjitwouldbedi culttohandthe nalexpressionforβn;

luckily,asweshallsee,theplanarlimit(N→∞)willenableustoneglectthedi erencesamongthesequantitiesrelativetodi erentilevels.InthislimitwemustcomputetheexpressionforβnsupposingthateachstepdownyieldsafactorR,andeachstepthatstaysatthesamelevelyieldsafactorA.Thusthequestionis:Howmanyarethestaircasesofistepswhose nale ectistogouponestep?LetjbethestepsoftypeA,thentheotheri jaredividedinpstepsupandp 1steps downsothati=j+2p 1.WithouttheAstepsthere2p 1arepstaircasesof2p 1stepswhose nale ectistogouponestep.InsidethesestaircaseswewanttoinserttheremainingjlevelsoftypeA:thereare2p places wheretheycanbeinserted,and,fora xed2p+j 1staircase,therearechoices.Finallythenumberofstaircasesj