Orthogonal polynomial method and odd vertices in matrix mode(2)

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.

2ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES

below,andthustheintegralgivingthepartitionfunctionwellde ned.Attheendonecantakethelimitwhenthecouplingconstantofthequarticvertexgoestozero.

Westartfromthepartitionfunction ZN(g)=dMe trS(M)(1)wheretheintegrationisoveranhermitianmatrixoforderNandwheretheactionisgiveningeneralby

1 S(M)=

jNj

Ni

ZN(0)=∞ h=0N2 2heh(g)(4)

where2 2histheEulercharacteristicoftheorientedribbongraphstobesummedintheperturbativeexpansionofthefunctionseh(g).Indeed,denotingsuchgraphswithcapitalletters,eachfunctionehadmitsthefollowingexpansion[1]

vi(G) igie

h

(g)=

Gconnectedofgenush

Anautomorphismofanorientedribbongraphisde nedinthefollowingway.Firstletusidentifytheorientedribbongraphasacommongraphplusacyclicorderingonthesetsofhalf-edgesattachedtoeachvertexandthende neanau-tomorphismoftheorientedribbongraphasanautomorphismofthegraphwhichleavestheorderingofeachvertexunchanged.It’sclearthattheautomorphismmustsendeachvertexintoavertexofequalvalence.2