Some Mixed-Moments of Gaussian Elliptic Matrices and Ginibre Matrices

TL;DR

This paper derives explicit formulas and asymptotic behaviors for mixed moments of Gaussian elliptic and Ginibre matrices, connecting combinatorial structures with matrix moment calculations to improve computational efficiency.

Contribution

It introduces a novel combinatorial approach linking non-crossing pairings and Temperley-Lieb diagrams to compute mixed moments efficiently.

Findings

Explicit formula for mixed moments of Gaussian elliptic matrices.

Polynomial-time algorithm for computing mixed moments.

Asymptotic analysis of mixed moments as matrix size grows.

Abstract

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.