斜正交多项式与部分对称实Ginibre系综
Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble
摘要：
部分对称的实Ginibre系综由实对称和实反对称高斯随机矩阵的线性组合构成。这样的矩阵通常具有实特征值和复特征值。对于固定数目的实特征值，早期的工作已经给出了联合特征值概率密度函数的显式形式。利用这一点，我们导出了相应的求和广义配分函数的Pfiran公式.这个Pfidian公式允许将恰好存在$k$特征值的概率写成具有显式项的行列式。它也可以用来给出关联函数的显式形式，只要计算了某些斜正交多项式。这个任务是完成厄米多项式，并允许我们继续分析各种缩放限制的相关性，包括其中的矩阵只有弱非对称。
英文摘要：
The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly $k$ eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly non-symmetric.
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