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 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.

17 pages, 15 figures

References in corpus (3)

• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices
• Large N_c confinement and turbulence
• Non-Hermitean Wishart random matrices (I)

Cited by in corpus (8)

• Spectral transitions and universal steady states in random Kraus maps and circuits
• Applications of Large Random Matrices in Communications Engineering
• Spectra of large time-lagged correlation matrices from Random Matrix Theory
• Spectral density of generalized Wishart matrices and free multiplicative convolution
• Complete diagrammatics of the single ring theorem
• Quaternionic R transform and non-hermitian random matrices
• Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices
• Summing free unitary Brownian motions with applications to quantum information