Multi-matrix models at general coupling

TL;DR

This paper analyzes the eigenvalue distributions of Hoppe's two and three matrix models across different coupling regimes, revealing transitions from perturbed semicircles to parabolic shapes and joint eigenvalue distributions.

Contribution

It provides a detailed analysis of eigenvalue distributions in multi-matrix models at various couplings and introduces techniques applicable to other multi-matrix models.

Findings

Eigenvalue distribution transitions from semicircle to parabola with increasing coupling

At large couplings, matrices approximately commute and joint eigenvalue distributions are derived

The methods developed are applicable to a broader class of multi-matrix models

Abstract

The eigenvalue distribution of Hoppe's two matrix model is investigated in detail as a function of the model's coupling. For small couplings it is a perturbed Wigner semicircle, while for large couplings it is a parabolic distribution which crosses over to a Wigner semicircle for eigenvalues within approximatley an inverse coupling from the boundary of the distribution. The model is approximately commuting at large couplings and we find the joint eigenvalue distribution of the two matrices. We also study a related three matrix model finding the corresponding three dimensional eigenvalue distribution there also. The techniques developed here are more widely applicable to other multi-matrix models.