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 $\mathbf{ X}_{ ji}^*$), where symbolically $\epsilon_i\in\{ 1,\dagger\}$, and where the expectation $\mathbb{ E}\left[\cdot\right]$ is taken over all matrices $\mathbf{ X}$. We start by finding an explicit formula for $\varphi(\mathbf{ X}^n,(\mathbf{ X}^\dagger)^m)$, $n,m\in\mathbb{ N}$, by using a mapping between non-crossing pairings on $\ell=n+m$ elements and Temperley-Lieb diagrams between two strands of $n$ and $m$ elements. This formula allows for a numerically efficient way to compute $\varphi(\mathbf{ X}^n,(\mathbf{ X}^\dagger)^m)$ 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 $n,m\to\infty$. 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.
Categories: math-ph math.CO math.MP math.PR