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.

中文翻译：

---

通过对称函数具有辛和正交不变性的乘积矩阵过程

我们应用对称函数理论来研究由Haar分布辛矩阵和正交矩阵截断乘积的奇异值形成的随机过程。这些产品矩阵过程是由 Borodin 和 Corwin 引入的 Macdonald 过程的退化。通过这种联系，我们获得了确定性矩阵的奇异值分布的显式公式，该公式乘以截断的 Haar 正交或辛矩阵，其中后者因素充当秩 $1$ 扰动。因此，我们将最近的 Kieburg-Kuijlaars-Stivigny 公式推广到截断酉矩阵乘积的联合奇异值密度到辛和正交对称类。专门研究两个辛矩阵的乘积，其秩为 $1$ 的微扰因子，