In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.

中文翻译：

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椭圆吉尼布矩阵实特征值的有限尺寸修正

在本文中，我们考虑正交对称类中的椭圆吉尼伯矩阵，它在实吉尼伯系综和高斯正交系综之间进行插值。我们在全局和边缘缩放机制以及强和弱非厄米性机制中获得了真实特征值密度的有限尺寸修正。我们的结果扩展并提供了在上述限制内先前最近发现的收敛率。特别是，在 Hermitian 极限下，我们的结果恢复了 Forrester、Frankel 和 Garoni 建立的高斯正交系综的有限尺寸修正。