论文标题
重新审视高斯正交和符号合奏的tracy-widom分布：一种偏斜 - 正交多项式方法
Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
论文作者
论文摘要
我们研究了随机矩阵理论的“ pfaffian”经典合奏中最大的特征值的分布，即在高斯正交（GOE）和高斯符号（GSE）的结合中，使用光学旋转的旋转旋转旋转式旋转式多核心，在nm and of theral and of nm of nm of nm of nm of nm for nm nm of nm of nm of nm sememials（GSE）。 （gue）。概括Adler，Forrester，Nagao和Van Moerbeke的技术，并使用Knuth引起的“重叠的Pfaffian”身份，我们以Nms nm nm nm nm in the the gue的nms研究的半古典式质量多种方向明确地构建了这些半古典偏斜 - 正交多发质。使用这些多项式，我们获得了GOE和GSE中最大特征值的累积分布函数的表达式。此外，通过对大型基质大小极限的这些偏斜 - 正交多项式进行渐近分析，我们获得了GOE和GSE的Tracy-Widom分布的替代推导。这种渐近分析依赖于某种Pfaffian身份，其证明在完美的匹配和链接图中采用了Pfaffians的表征。
We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.
