Summing free unitary random matrices

TL;DR

This paper develops quaternion free probability calculus to analyze the eigenvalue and singular value distributions of sums of independent unitary matrices, providing new theoretical results and numerical solutions for specific cases.

Contribution

It introduces quaternion free probability calculus for non-Hermitian matrices and derives new large-size limit formulas for sums of unitary matrices, including master equations and conjectures.

Findings

Derived mean eigenvalue and singular value densities for sums of unitary matrices

Solved master equations for CUE summands and tested numerically

Proved a central limit theorem and first correction for i.i.d. zero-drift unitary matrices

Abstract

I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two "master equations," which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit theorem, and its first sub-leading correction, for independent identically-distributed zero-drift unitary random matrices.