Product matrix processes as limits of random plane partitions

TL;DR

This paper studies a discrete-time random process involving singular values of products of truncated Haar unitary matrices, revealing it as a limit of the Schur process and connecting it to random plane partitions.

Contribution

It establishes a scaling limit of the Schur process for these matrix products and links continuous marginals of q-distributed plane partitions to singular value distributions.

Findings

Determinantal formulas for correlation functions

Contour integral representation of the correlation kernel

Connection between plane partitions and products of random matrices

Abstract

We consider a random process with discrete time formed by singular values of products of truncations of Haar distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.