Gaussian diagrammatics from Circular Ensembles of random matrices

TL;DR

This paper reveals a hidden Gaussian structure within circular ensembles of random matrices, enabling new diagrammatic methods for calculating moments, with applications to moments of submatrix traces.

Contribution

It introduces a novel Gaussian ensemble inside each circular ensemble, providing new diagrammatic rules for moment calculations across different symmetry classes.

Findings

Identifies a hidden Gaussian ensemble in circular ensembles

Develops new diagrammatic rules for moments

Calculates moments of traces of submatrices

Abstract

We uncover a hidden Gaussian ensemble inside each of the three circular ensembles of random matrices, which provide novel diagrammatic rules for the calculation of moments. The matrices involved are generic complex for $\beta=2$, complex symmetric for $\beta=1$ and complex self-dual for $\beta=4$, and their dimension must be set to $1-2/\beta$. As an application, we compute moments of traces of submatrices.