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

TL;DR

This paper develops a perturbative framework to analyze the critical behavior of two-matrix models in the large N limit, classifying universality classes based on the singularity structure of associated differential operators.

Contribution

It introduces a novel perturbative approach to characterize critical points and universality classes in two-matrix models using high-order expansions of the Heisenberg algebra representations.

Findings

Classifies universality classes as [p,q] based on operator singularities.

Distinguishes cases where $l_2$ divides $l_1$ and where it does not.

Provides a systematic method for analyzing critical behavior in matrix models.

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.