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.
翻译:我们显示,近年来引入的对称空间的Riemannian Gaussian分布是标准的随机矩阵类型。 我们利用它来对概率密度函数的边际进行分析计算。 可以用Stieltjes- Wigert 或thogonal 多元数学来进行充分计算。 就Hermitian 矩阵的空间而言, 分布已经出现在物理文献中。 对于对称空间是 $m\ times mayme mayme symal 确定矩阵的空间的情况, 我们展示了如何通过对 Pfaffians 进行特定值的 $m$ 来有效计算。 相当地, 我们也可以通过建立特定的正正数矩阵矩阵矩阵空间(skew Stieltjes-Wigert 多边数学中已经出现) 来取得相同的结果 。 对于其他对称空间是 $mydnic case case, 我们展示了概率的密度函数是如何以特定的例子, 也就是在 Exmissional- cal- cal rog cal 中, 我们也可以制模制模制模的磁盘中, 。