论文标题
Riemannian高斯分布，随机矩阵合奏和扩散内核
Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
论文作者
论文摘要
我们表明，近年来引入的对称空间上的Riemannian高斯分布是标准的随机矩阵类型。我们利用它来计算概率密度函数的分析边缘。可以使用stieltjes-wigert正交多项式来充分完成此操作，对于赫尔米尼矩阵的空间，分布已经出现在物理文献中。对于对称空间是$ M \ times m $对称正定矩阵的空间时，我们展示了如何通过以$ m $的特定值评估pfaffians来有效地计算。同等地，我们可以通过构建特定的偏斜正交多项式来获得相同的结果，以进行对数正态重量函数（偏斜stieltjes-wigert多项式）。研究了其他对称空间，并在Quaternionic情况下获得了相同类型的结果。此外，我们展示了概率密度函数如何是扩散繁殖Karlin-McGregor类型内核的特殊情况，描述了非互相关的Brownian Motions，这也是Lie基团的Weyl combern中的扩散过程。
We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done fully, using Stieltjes-Wigert orthogonal polynomials, for the case of the space of Hermitian matrices, where the distributions have already appeared in the physics literature. For the case when the symmetric space is the space of $m \times m$ symmetric positive definite matrices, we show how to efficiently compute by evaluating Pfaffians at specific values of $m$. Equivalently, we can obtain the same result by constructing specific skew orthogonal polynomials with regards to the log-normal weight function (skew Stieltjes-Wigert polynomials). Other symmetric spaces are studied and the same type of result is obtained for the quaternionic case. Moreover, we show how the probability density functions are a particular case of diffusion reproducing kernels of the Karlin-McGregor type, describing non-intersecting Brownian motions, which are also diffusion processes in the Weyl chamber of Lie groups.
