反对称化与厄米化矩阵乘积系综的组合学与圈方程
Combinatorics and loop equations for antisymmetrised and Hermitised matrix product ensembles
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中文总结 AI 辅助
本研究针对反对称化与厄米化矩阵乘积系综，构造由混合累积量计数的带状图和星座图，并在m=1时给出生成函数的圈方程刻画。
中文摘要 AI 辅助
$m$ 阶反对称化矩阵乘积系综由乘积 $X_1^T\cdots X_m^TJX_m\cdots X_1$ 表示，其中 $X_1,\ldots,X_m$ 是独立的实 Ginibre 矩阵，$J$ 是基本反对称矩阵；而 $m$ 阶厄米化矩阵乘积系综由乘积 $X_1^\dagger\cdots X_m^\dagger HX_1\cdots X_m$ 表示，其中 $X_1,\ldots,X_m$ 现在是独立的复 Ginibre 矩阵，$H$ 是从高斯酉系综中抽取的厄米矩阵。这些系综最近已被证明与某些 Muttalib--Borodin 系综及 Harish-Chandra--Itzykson--Zuber 型积分相关，从而激发了对它们特征值统计的进一步研究。在本工作中，我们构造了由这些系综的混合累积量计数的带状图与星座图，并在 $m=1$ 时给出了所述累积量生成函数的圈方程刻画。
英文摘要
The order $m$ antisymmetrised matrix product ensemble is represented by the product $X_1^T\cdots X_m^TJX_m\cdots X_1$, where $X_1,\ldots,X_m$ are independent real Ginibre matrices and $J$ is the elementary antisymmetric matrix, while the order $m$ Hermitised matrix product ensemble is represented by $X_1^\dagger\cdots X_m^\dagger HX_1\cdots X_m$, where $X_1,\ldots,X_m$ are now independent complex Ginibre matrices and $H$ is a Hermitian matrix drawn from the Gaussian unitary ensemble. These ensembles have recently been shown to be related to certain Muttalib--Borodin ensembles and integrals of Harish-Chandra--Itzykson--Zuber type, thereby motivating further investigation into their eigenvalue statistics. In this work, we construct ribbon graphs and constellations that are enumerated by the mixed cumulants of these ensembles and give loop equation characterisations for the generating functions of said cumulants when $m=1$.
发表机构
- Université de Bordeaux（波尔多大学）
- The University of Hong Kong（香港大学）
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