On the limiting Horn inequalities

The Horn inequalities characterise the possible spectra of triples of $n$-by-$n$ Hermitian matrices $A+B=C$. We study integral inequalities that arise as limits of Horn inequalities as $n \to \infty$. These inequalities are parametrised by the points of an infinite-dimensional convex body, the asymptotic Horn system $\mathscr{H}[0,1]$, which can be regarded as a topological closure of the countable set of Horn inequalities for all finite $n$. We prove three main results. The first shows that arbitrary points of $\mathscr{H}[0,1]$ can be well approximated by specific sets of finite-dimensional Horn inequalities. Our second main result shows that $\mathscr{H}[0,1]$ has a remarkable self-characterisation property. That is, membership in $\mathscr{H}[0,1]$ is determined by the very inequalities corresponding to the points of $\mathscr{H}[0,1]$ itself. To illuminate this phenomenon, we sketch a general theory of sets that characterise themselves in the sense that they parametrise their own membership criteria, and we consider the question of what further information would be needed in order for this self-characterisation property to determine the Horn inequalities uniquely. Our third main result is a quantitative result on the redundancy of the Horn inequalities in an infinite-dimensional setting. Concretely, the Horn inequalities for finite $n$ are indexed by certain sets $T^n_r$ with $1 \le r \le n-1$; we show that if $(n_k)_{k \ge 1}$ and $(r_k)_{k \ge 1}$ are any sequences such that $(r_k / n_k)_{k \ge 1}$ is a dense subset of $(0,1)$, then the Horn inequalities indexed by the sets $T^{n_k}_{r_k}$ are sufficient to imply all of the others.