Some Mixed-Moments of Gaussian Elliptic Matrices and Ginibre Matrices

We consider the mixed-moments $\phi ({\mathbf{X}}^{{\epsilon }_1},\dots ,{\mathbf{X}}^{{\epsilon }_k})={\lim }_{N\to \mathrm{\infty }}{N}^{-1}\mathbb{E}\left[\mathrm{T}\mathrm{r}({\mathbf{X}}_1^{\epsilon }\cdots {\mathbf{X}}^{{\epsilon }_k})\right]$ of complex Gaussian Elliptic Matrices $\mathbf{X}$ (with correlation parameter $\rho$ between elements ${\mathbf{X}}_{ij}$ and ${\mathbf{X}}_{ji}^{\ast }$), where symbolically ${\epsilon }_i\in \{1,†\}$, and where the expectation $\mathbb{E}[\cdot ]$ is taken over all matrices $\mathbf{X}$. We start by finding an explicit formula for $\phi ({\mathbf{X}}^n,({\mathbf{X}}^{†}{)}^m)$, $n,m\in \mathbb{N}$, by using a mapping between non-crossing pairings on $\mathrm{\ell }=n+m$ elements and Temperley-Lieb diagrams between two strands of $n$ and $m$ elements. This formula allows for a numerically efficient way to compute $\phi ({\mathbf{X}}^n,({\mathbf{X}}^{†}{)}^m)$ by reducing the exponential complexity of a naive enumeration of non-crossing pairings to polynomial complexity. We also provide the asymptotic behavior of these mixed-moments as $n,m\to \mathrm{\infty }$. We then provide an explicit computation for some more general mixed-moments by considering the position of the matrix $\mathbf{X}$ in the product ${\mathbf{X}}^{{\epsilon }_1}\cdots {\mathbf{X}}^{{\epsilon }_k}$. We, therefore, deduce closed-form formulas for some mixed-moments of Ginibre matrices.