Product Matrix Processes with Symplectic and Orthogonal Invariance via Symmetric Functions

TL;DR

This paper uses symmetric function theory to analyze singular values of products involving truncated Haar symplectic and orthogonal matrices, extending existing formulas to new symmetry classes and revealing Pfaffian point process structures.

Contribution

It generalizes the Kieburg-Kuijlaars-Stivigny formula to symplectic and orthogonal cases and connects product matrix processes to Macdonald processes.

Findings

Explicit distribution formulas for singular values under rank 1 perturbations.

Extension of joint singular value density formulas to symplectic and orthogonal classes.

Squared singular values form a Pfaffian point process in specific cases.

Abstract

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.