A note on the tensor product of two random unitary matrices

TL;DR

This paper studies the eigenvalue distribution of tensor products of independent random unitary matrices, revealing a transition from superimposed sine processes to a Poisson process as matrix sizes grow.

Contribution

It characterizes the limiting eigenvalue point processes of tensor products of random unitary matrices as their dimensions increase.

Findings

Eigenvalues form a superposition of m sine processes for large n.

As both m and n grow, the eigenvalue process converges to a Poisson process.

Provides insight into spectral behavior of tensor products of random matrices.

Abstract

In this note we consider the point process of eigenvalues of the tensor product of two independent random unitary matrices of size m by m and n by n. When n becomes large, the process behaves like the superposition of m independent sine processes. When m and n go to infinity, we obtain the Poisson point process in the limit.