Renormalization Group Approach to Matrix Models and Vector Models

TL;DR

This paper explores the renormalization group method for large N matrix and vector models, clarifying its validity and providing exact fixed point and flow analyses for these models.

Contribution

It demonstrates the validity of the RG approach for matrix models using vector models as a simpler example and derives exact flow equations and fixed points.

Findings

Exact difference equations for free energies at different N

Reparametrization reduces infinite coupling space to finite dimensions

Detailed RG flow diagrams for models with up to two couplings

Abstract

The renormalization group approach is studied for large $N$ models. The approach of Br\'ezin and Zinn-Justin is explained and examined for matrix models. The validity of the approach is clarified by using the vector model as a similar and simpler example. An exact difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space provides infinitely many identities which reduce the infinite dimensional coupling constant space to that of finite dimensions. The effective beta functions give exact values for the fixed points and the susceptibility exponents. A detailed study of the effective renormalization group flow is presented for cases with up to two coupling constants. We draw the two-dimensional flow diagram.