Gaussian diagrammatics from Circular Ensembles of random matrices

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 $1-2\mathrm{/}\beta$. As an application, we compute moments of traces of submatrices.