Summing free unitary random matrices.

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 circular unitary ensemble 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 subleading correction, for independent identically distributed zero-drift unitary random matrices.