论文标题
高斯椭圆矩阵和吉尼伯矩阵的一些混合词
Some Mixed-Moments of Gaussian Elliptic Matrices and Ginibre Matrices
论文作者
论文摘要
我们考虑混合词$φ（\ mathbf {x}^{ε_1}，\ ldots，\ mathbf {x}}^{ε_k}）= \ lim_ {n \ to \ infty} n^{ - 1} e} \ left [\ mathrm {tr} \ left（\ mathbf {x}^ε_1\ cdots \ cdots \ mathbf {x}^{ε_k} \ right）\ right] $复杂的高斯椭圆矩阵$ \ mathbf {x} $（具有相关参数$ρ$之间的元素$ \ mathbf {x} _ {ij} $和$ \ mathbf {x} _ {x} _ {ji}^$） $ \ mathbb {e} \ left [\ cdot \ right] $均在所有矩阵$ \ mathbf {x} $上。我们首先要找到$φ（\ mathbf {x}^n，（\ Mathbf {x}}^\ dagger）^m）$，$ n，m \ in \ mathbb {n} $的明确公式，（元素。该公式允许一种有效的方法来计算$φ（\ mathbf {x}^n，（\ mathbf {x}}^\ dagger）^m）$，通过降低非跨配对配对的幼稚枚举的指数复杂性，从而降低非跨配对的指数复杂性。我们还提供了这些混合词的渐近行为，即$ n，m \至\ infty $。然后，我们通过考虑矩阵$ \ mathbf {x} $在产品中的位置$ \ mathbf {x}^{x}^{ε_1} \ cdots \ cdots \ mathbf {x}^{X}^{ε_k} $，为一些更通用的混合词提供明确的计算。因此，我们推断出一些吉尼伯矩阵混合词的封闭式公式。
We consider the mixed-moments $φ(\mathbf{X}^{ε_1},\ldots,\mathbf{X}^{ε_k})=\lim_{N\to\infty}N^{-1}\mathbb{E}\left[\mathrm{Tr}\left(\mathbf{X}^ε_1\cdots\mathbf{X}^{ε_k}\right)\right]$ of complex Gaussian Elliptic Matrices $\mathbf{X}$ (with correlation parameter $ρ$ between elements $\mathbf{X}_{ij}$ and $\mathbf{X}_{ji}^*$), where symbolically $ε_i\in\{1,\dagger\}$, and where the expectation $\mathbb{E}\left[\cdot\right]$ is taken over all matrices $\mathbf{X}$. We start by finding an explicit formula for $φ(\mathbf{X}^n,(\mathbf{X}^\dagger)^m)$, $n,m\in\mathbb{N}$, 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 $φ(\mathbf{X}^n,(\mathbf{X}^\dagger)^m)$ 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 $\mathbf{X}^{ε_1}\cdots\mathbf{X}^{ε_k}$. We, therefore, deduce closed-form formulas for some mixed-moments of Ginibre matrices.
