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The Ginibre ensembles are non-Hermitian random matrices with real, complex or quaternion Gaussian entries. For reasons to be discussed below, these are denoted by GinOE, GinUE and GinSE respectively. With the quaternion entries of the latter identified as the \(2 \times 2\) complex block structure \( \begin{bmatrix} z & w \\ - \bar{w} & \bar{z} \end{bmatrix}, \) members of each of these ensembles are readily realised; Fig. 1.1 gives example plots of the corresponding eigenvalue distribution in the complex plane. Ginibre introduced these non-Hermitian random matrix ensembles in 1965 [293]. Although with a self-described motivation of mathematical curiosity [31, Sect. 2.2, quoting correspondence with Ginibre], the line of investigation follows on from ideas and methods put forward by Dyson a few years earlier.

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Introduction

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  • 出版模式：

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  • 发表日期：

    2025年

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  • Sung-Soo Byun,
  • Peter J. Forrester

摘要

The Ginibre ensembles are non-Hermitian random matrices with real, complex or quaternion Gaussian entries. For reasons to be discussed below, these are denoted by GinOE, GinUE and GinSE respectively. With the quaternion entries of the latter identified as the \(2 \times 2\) complex block structure \( \begin{bmatrix} z & w \\ - \bar{w} & \bar{z} \end{bmatrix}, \) members of each of these ensembles are readily realised; Fig. 1.1 gives example plots of the corresponding eigenvalue distribution in the complex plane. Ginibre introduced these non-Hermitian random matrix ensembles in 1965 [293]. Although with a self-described motivation of mathematical curiosity [31, Sect. 2.2, quoting correspondence with Ginibre], the line of investigation follows on from ideas and methods put forward by Dyson a few years earlier.