Perturbative approach to the critical behaviour of two-matrix models in the limit N -> infinity

Abstract

We construct representations of the Heisenberg algebra by pushing the perturbation expansion to high orders. If the multiplication operators $B_{1,2}$ tend to differential operators of order $l_{2,1}$, respectively, the singularity is characterized by $(l _{1},l_{2})$. Let $l_{1} \geq l_{2}$. Then the two cases A : ``$l_{2}$ does not divide $l_{1}$'' and B : ``$l_{2}$ divides $l_{1}$'' need a different treatment. The universality classes are labelled $[p,q]$ where $[p,q]$=[$l_{1}$,$l_{2}$] in case A and $[p,q]$=[$l_{1}+1$,$l_{2}$] in case B.