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We consider the mixed-moments$\varphi(\mathbf{X}^{\epsilon_1},\ldots,\mathbf{X}^{\epsilon_k})=\lim_{N\to\infty}N^{-1}\mathbb{E}\left[\mathrm{Tr}\left(\mathbf{X}^\epsilon_1\cdots\mathbf{X}^{\epsilon_k}\right)\right]$of complex Gaussian Elliptic Matrices $\mathbf{X}$ (with correlation parameter$\rho$ between elements $\mathbf{X}_{ij}$ and $\mathbf{X}_{ji}^*$), wheresymbolically $\epsilon_i\in\{1,\dagger\}$, and where the expectation$\mathbb{E}\left[\cdot\right]$ is taken over all matrices $\mathbf{X}$. Westart by finding an explicit formula for$\varphi(\mathbf{X}^n,(\mathbf{X}^\dagger)^m)$, $n,m\in\mathbb{N}$, by using amapping between non-crossing pairings on $\ell=n+m$ elements and Temperley-Liebdiagrams between two strands of $n$ and $m$ elements. This formula allows for anumerically efficient way to compute$\varphi(\mathbf{X}^n,(\mathbf{X}^\dagger)^m)$ by reducing the exponentialcomplexity of a naive enumeration of non-crossing pairings to polynomialcomplexity. We also provide the asymptotic behavior of these mixed-moments as$n,m\to\infty$. We then provide an explicit computation for some more generalmixed-moments by considering the position of the matrix $\mathbf{X}$ in theproduct $\mathbf{X}^{\epsilon_1}\cdots\mathbf{X}^{\epsilon_k}$. We, therefore,deduce closed-form formulas for some mixed-moments of Ginibre matrices.

Some Mixed-Moments of Gaussian Elliptic Matrices and Ginibre Matrices