title: Summing free unitary random 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 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.
authors.affiliation: Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, 31-342 Kraków, Poland. jedrekjarosz@gmail.com
authors.forename: Andrzej
authors.initials: A
authors.lastname: Jarosz
mesh_terms: D000465:Algorithms; D001703:Biophysics / Q000379:methods*; D015233:Models, Statistical; D008962:Models, Theoretical; D011336:Probability; D011789:Quantum Theory; D015203:Reproducibility of Results; D013816:Thermodynamics