Horn's spectrum problem with random Hermitian matrices

Question

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 $\vec b$ ?

Answer 1

The unitary group $U(n)$ acts on the space $X$ of all $n\times n$ matrices with fixed eigenvalues (and multiplicities, to be precise) by conjugation and this action is transitive. As a result there exists a unique probability measure on $X$ which is invariant under unitary conjugation. Sampling that measure is equivalent to sampling $U A U^*$ with $U$ distributed according to the Haar measure and $A\in X$ a fixed Hermitian matrix.

The asymptotic question is well-understood through the connection with Voiculescu's free probability theory. It is known that (under the assumptions that you make and also under the additional assumption that your measures $\mu_n$ and $\nu_n$ are supported on some compact set $K$ for all $n$) that the spectral measure of $A+B$ almost surely converges to the free additive convolution $\eta = \mu_\infty \boxplus \nu_\infty$. Furthermore, using Voiculescu's $R$-transform, it is possible to compute the Stieltjes transform of $\eta$ in terms of Stieltjes transforms for $\mu_\infty$ and $\nu_\infty$ (what is involved is inverting some analytic functions). In fact, since the matrices $A$ and $B$ become asymptotically freely independent, there is (at least in principle) a way to answer similar question for more complicated expressions (e.g. products, commutators).

The behavior for finite $n$ is also of significant interest. A very recent result of Marcus, Speilman and Srivastava computes precisely the expected value of the characteristic polynomial of $A+B$ in terms of those of $A$ and $B$. This does not quite give what you might want, but gives amazing control on spectral properties of the sum.