An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc., 12 (1999), pp 1055–1090), is

Given two $n\times n$ Hermitian matrices $A$ and $B$, describe all the possible spectra of $A+U^* BU$, as $U$ runs over the unitary group.

The answer was conjectured by A. Horn (who is distinct from R. Horn, the co-author of the book with Johnson) in 1962. The possible spectra form in ${\mathbb R}^n$ a polytope defined by a rather intricate list of linear inequalities. The first ones were derived by H. Weyl (1912). Then came Ky Fan's inequalities (1949) and Wielandt's ones (1955).

My question is of a different nature:

What is the distribution of the spectrum of $A+U^* BU$ when $U^* BU$ is taken randomly among the matrices with fixed prescribed spectrum ?

By the way, what is a natural probabilistic framework ? One could take $U$ randomly, according to the Haar measure, but I do not see a clear argument for that.

Also, because the answers to questions about random matrices are often easier when the size $n$ tends to infinity, let me ask the following.

For $n>\!\!>1$, let us prescribe two real spectra $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ (multiplicities are allowed). Suppose that the empirical measures $$\mu_n=\frac1n\sum_1^n\delta_{a_j},\qquad\nu_n=\frac1n\sum_1^n\delta_{b_j}$$ converge to probabilities $\mu_\infty$ and $\nu_\infty$. How does the spectrum of $A+B$ behave asymptotically, when the Hermitian matrices $A$ and $B$ are chosen randomly with prescribed spectra $\vec a$ and $\vec b$ ?