吉尼布尔系综II：GinOE和GinSE的研究进展
Progress on the study of the Ginibre ensembles II: GinOE and GinSE
摘要：
这是第二部分的审查有关的三个类的随机非厄米高斯矩阵介绍了吉尼布尔于1965年。虽然第一部分限制注意GinUE（Ginibre酉系综）的情况下，复杂的元素，在这一部分的情况下，真正的元素（GinOE，表示Ginibre正交系综）和四元数元素表示为$2 \times 2$复杂的块（GinSE，表示Ginibre辛系综）考虑。GinOE和GinSE的特征值都形成普法菲点过程，该过程比GinUE产生的行列式点过程更复杂。尽管如此，许多阻碍理论传统方面发展的障碍现在已经克服，而新的理论方面和新的应用已经确定。这允许一个全面的帐户的主题处理太复杂的情况下：本征值概率密度函数和关联函数，限制公式的关联函数，波动公式，求和规则，间隙概率和特征向量统计，等等。与复杂情况不同的是，需要发展一种与Pfidian相关的斜内积相对应的斜正交多项式理论。另一个不同的主题是统计的实际特征值，这是唯一的Ginoe。这些出现在一些应用的理论，来自不同的领域，如扩散过程和统计物理学的持久性，拓扑驱动的参数能级交叉某些量子点，和平衡计数系统的随机非线性微分方程。
英文摘要：
This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as $2 \times 2$ complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.
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