实Ginibre系综的圆盘计数统计
Disc counting statistics of the real Ginibre ensemble
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中文总结 AI 辅助
该研究推导实Ginibre系综不同区域圆盘内实与非实特征值数量的联合累积量渐近行为，建立联合中心极限定理，揭示其联合涨落的普适性现象。
中文摘要 AI 辅助
我们研究实Ginibre系综的圆盘计数统计，其谱包含实特征值与非实特征值，非实特征值以复共轭对形式出现。随着矩阵维度增大，我们推导了圆心在圆律的体区、边区及外区的圆盘内，实特征值与非实特征值数量的联合累积量的渐近行为。作为推论，我们建立了联合中心极限定理，并得到了数方差的猜想渐近式。此外，我们的结果揭示了一种普适性现象：尽管实特征值与非实特征值各自的涨落仍保留对对称类的依赖，它们的联合涨落却呈现出与此前已确立的复Ginibre系综和辛Ginibre系综相同的极限行为。
英文摘要
We study the disc counting statistics of the real Ginibre ensemble, whose spectrum consists of real and non-real eigenvalues, the latter occurring in complex-conjugate pairs. As the matrix dimension increases, we derive the asymptotic behaviour of the joint cumulants of the numbers of real and non-real eigenvalues contained in a centred disc in the bulk, edge, and exterior regimes of the circular law. As consequences, we establish a joint central limit theorem and obtain the conjectured asymptotics for the number variance. Furthermore, our results reveal a universality phenomenon: although the fluctuations of the real and non-real eigenvalues separately retain dependence on the symmetry class, their combined fluctuations exhibit the same limiting behaviour as in the previously established complex and symplectic Ginibre ensembles.
