Orthogonal polynomial method and odd vertices in matrix mode(10)

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.

ORTHOGONALPOLYNOMIALMETHODANDODDVERTICES11su ces.Thequarticcasecanbeexplicitlyintegrated[1]toobtain

∞ 1(2k 1)!e

(g4)=(3g4)k(a 1)(a 9)=24k=1

1 12g4

(k+1)!(k 1)!.(41)

Recallingtheformulaforthetopologicalexpansion(5),onehastheinterestingequation

1(42)k!(k+2)!Gplanar,connected,withkquarticvertices

thatcanbecheckedfork=1andk=2usingthecontentsof gure2.Theradiusofconvergenceis1/12andg4c=1/12isthecriticalpoint.Forg4→g4coneobtainsthecriticalbehavior

e0(g4)～(g4c g4)5

=1 2g3AR

2g3R=A g3A2.

22g3x+σ(1+σ)(1+2σ)=0.(44)Infact,letusintroducethenewvariableσ= g3Arelatedtoxby(45)

2Thefunctionσ(x)=σ¯(g3x)isthesolutionof(45)whichvanishesfor

x=0;indeedwheng3=0thepotentialhasnolongeroddverticesandthenA(x)=0 σ¯(0)=0.Ourfunctione0(g3)mayberewritten,goingoverfromthevariablextothevariableσandintegratingbyparts

e0(g3)= 1

dx(1 x)ln

10 R(x)1+2σ(46)

= 3(1+σ1)(1+2σ1)2