论文标题
通过对称函数具有符号和正交不变性的产品矩阵过程
Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions
论文作者
论文摘要
我们将对称函数理论应用于研究由HAAR分布式符号和正交矩阵截断产物的奇异值形成的随机过程。这些产品矩阵过程是Borodin和Corwin引入的MacDonald过程的退化。通过这种连接，我们获得了确定性矩阵的单数值的分布乘以截短的HAAR正交或符号矩阵的明确公式，在后一个因子充当等级$ 1 $扰动的条件下。因此，我们将最近的Kieburg-Kuijlaars-Stivigny公式推广到截短单位矩阵的产物的联合奇异值密度为符号和正交对称类别。专门针对两个具有等级$ 1 $扰动因子的符号矩阵的产品，我们表明平方奇异值构成了PFAFFIAN点过程。
We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald processes introduced by Borodin and Corwin. Through this connection, we obtain explicit formulae for the distribution of singular values of a deterministic matrix multiplied by a truncated Haar orthogonal or symplectic matrix under conditions where the latter factor acts as a rank $1$ perturbation. Consequently, we generalize the recent Kieburg-Kuijlaars-Stivigny formula for the joint singular value density of a product of truncated unitary matrices to symplectic and orthogonal symmetry classes. Specializing to products of two symplectic matrices with a rank $1$ perturbative factor, we show that the squared singular values form a Pfaffian point process.
