On the Large N Limit of 3D and 4D Hermitian Matrix Models

TL;DR

This paper investigates the large N limit of hermitian matrix models in three and four dimensions using an approximate Renormalization Group approach, revealing a non-trivial fixed point in 3D but not in 4D.

Contribution

It introduces an approximate RG method to analyze the large N limit of matrix models in 3D and 4D, identifying fixed points and critical exponents, highlighting universality.

Findings

In 4D, the model lacks an interacting continuum limit.

In 3D, a non-trivial fixed point exists with specific critical exponents.

Critical exponents are largely independent of approximation details.

Abstract

The large N limit of the hermitian matrix model in three and four Euclidean space-time dimensions is studied with the help of the approximate Renormalization Group recursion formula. The planar graphs contributing to wave function, mass and coupling constant renormalization are identified and summed in this approximation. In four dimensions the model fails to have an interacting continuum limit, but in three dimensions there is a non trivial fixed point for the approximate RG relations. The critical exponents of the three dimensional model at this fixed point are $\nu = 0.665069$ and $\eta=0.19882$. The existence (or non existence) of the fixed point and the critical exponents display a fairly high degree of universality since they do not seem to depend on the specific (non universal) assumptions made in the approximation.