椭圆GinOE矩阵实特征值涨落的局部中心极限定理
Local central limit theorem for real eigenvalue fluctuations of elliptic GinOE matrices
AI总结：
本研究针对椭圆GinOE矩阵实特征值数量的涨落，通过证明生成函数仅有负实零点并利用方差发散性，将中心极限定理加强为局部中心极限定理，并讨论了收敛速率。
摘要：
椭圆Ginibre正交系综（GinOE）中的随机矩阵是实对称与实反对称实高斯随机矩阵的某种线性组合，由参数$τ$控制。我们关注实特征值数量的涨落，当期望数量与矩阵大小的平方根$N$成正比时，固定$τ$；以及当$τ$缩放至弱非对称极限时，特征值数量与$N$成正比。通过证明确定实特征值数量分布的生成函数仅有负实零点，并利用两种感兴趣情形下方差随$N \to \infty$趋于无穷的事实，将已知的涨落中心极限定理加强为局部中心极限定理，并讨论了收敛速率。
英文摘要：
Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a certain linear combination of a real symmetric, and real anti-symmetric, real Gaussian random matrices and controlled by a parameter $τ$. Our interest is in the fluctuations of the number of real eigenvalues, for fixed $τ$ when the expected number is proportional to the square root of the matrix size $N$, and for $τ$ scaled to the weakly non-symmetric limit, when the number of eigenvalues is proportional to $N$. By establishing that the generating function for the probabilities specifying the distribution of the number of real eigenvalues has only negative real zeros, and using too the fact that variances in both circumstancesof interest tends to infinity as $N \to \infty$, the known central limit theorem for the fluctuations is strengthened to a local central limit theorem, and the rate of convergence is discussed.
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