Spiked Models in Wishart Ensemble
Spiked Models in Wishart Ensemble
复制标题
威沙特合奏中的尖刺模型
DOI：
--
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
Dong Wang
中科院分区：
文献类型：
--
作者：
Dong Wang
尖刺模型是 Wishart 系综的一个重要特例，也是白色 Wishart 系综的自然概括。在数学上，它可以定义为三种变量：实数、复数和四元数。对于实际应用，我们感兴趣的是最大样本特征值的极限分布。
我们首先基于 Bleher 和 Kuijlaars 的多重正交多项式方法，对 Baik、Ben Arous 和 P'{e}ch'{e} 对于复杂尖峰模型的结果给出了新的证明。然后，本着同样的精神，我们提出了 1 阶四元尖峰模型的新结果，通过涉及四元数区域多项式（alpha = 1/2 Jack 多项式）和倾斜正交多项式的组合恒等式来证明。
我们发现随着尖峰总体特征值的增加，1 阶四元尖峰模型中的极限分布出现相变现象，并将临界点上看似新的极限分布识别为真实白 Wishart 系综中最大样本特征值的极限分布。
最后我们给出了高阶四元尖峰模型和真实尖峰模型的猜想。
The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For practical application, we are interested in the limiting distribution of the largest sample eigenvalue.
We first give a new proof of the result of Baik, Ben Arous and P'{e}ch'{e} for the complex spiked model, based on the method of multiple orthogonal polynomials by Bleher and Kuijlaars. Then in the same spirit we present a new result of the rank 1 quaternionic spiked model, proven by combinatorial identities involving quaternionic Zonal polynomials (alpha = 1/2 Jack polynomials) and skew orthogonal polynomials.
We find a phase transition phenomenon for the limiting distribution in the rank 1 quaternionic spiked model as the spiked population eigenvalue increases, and recognize the seemingly new limiting distribution on the critical point as the limiting distribution of the largest sample eigenvalue in the real white Wishart ensemble.
Finally we give conjectures for higher rank quaternionic spiked model and the real spiked model.