2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals

TL;DR

This paper demonstrates that 2-Hermitian and complex matrix models share the same loop equations, and introduces a new, efficient formula for unitary group integrals by expressing them as Gaussian integrals over triangular matrices.

Contribution

It establishes the equivalence of loop equations for two matrix models and derives a novel explicit formula for unitary integrals as triangular integrals.

Findings

2-Hermitian and complex matrix models obey the same loop equations

Derived an explicit formula for Itzykson-Zuber integrals over the unitary group

Expressed integrals over U(n) as Gaussian integrals over triangular matrices

Abstract

We prove that the 2-hermitean matrix model and the complex-matrix model obey the same loop equations, and as a byproduct, we find a formula for Itzykzon-Zuber's type integrals over the unitary group. Integrals over U(n) are rewritten as gaussian integrals over triangular matrices and then computed explicitly. That formula is an efficient alternative to the former Shatashvili's formula.