On the limiting Horn inequalities

TL;DR

This work studies the limiting behavior of Horn inequalities for triples of Hermitian matrices by introducing the asymptotic Horn system $\mathscr{H}[0,1]$, a topological closure capturing integral inequalities on triples of probability measures. It develops a quantile-function framework that recasts finite-n Horn data as points in an infinite-dimensional space, enabling three main results: (i) any element of $\mathscr{H}[0,1]$ can be approximated by finite-dimensional Horn data with prescribed limiting ratios; (ii) a self-characterisation property shows membership in the asymptotic systems is determined by inequalities indexed by the system itself; (iii) a quantitative redundancy result demonstrates that certain dense-sequence subsets of Horn inequalities suffice to imply the full infinite system. These findings reveal deep structural features, including invariances, convexity properties, and connections to free probability, and they open directions toward uniqueness questions and links to broader areas in operator theory and symplectic geometry. The combined approach solidifies understanding of infinite-dimensional Horn-type problems and their potential applications in random matrices and related fields.

Abstract

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.

Figures (3)

• Figure 1: Three $n$-integral and $r$-atomic quantile functions with $n=12$ and $r = 5$. These are associated with the Horn triple I = \{ 1, 2, 4, 5, 8 \} , \quad J = \{ 2,4,6,8,11 \} \quad \text{and} \quad K = \{ 2, 4, 7, 11, 12 \}.Lemma \ref{['lem:embed']} states that the $n$-integral and $r$-atomic quantile functions lying in $\mathscr{H}[0,1-r/n]$ are in bijection with the sets $T_r^n$ indexing the Horn inequalities.
• Figure 2: A relation on the set $X = \{x_1,\ldots,x_5\}$. We draw a green arrow from $x_i$ to $x_j$ if $x_i$ likes $x_j$. The subset $A = \{x_1,x_2,x_3\}$ is strongly self-characterising, as all elements of $A$ like each other and themselves, and every element not in $A$ dislikes an element of $A$.
• Figure 3: A relation on the set $X = \{x,y\}$ for which there are no strongly self-characterising sets. (A green arrow from $a$ to $b$ denotes $aLb$.) The friendly sets are the empty set and the singleton $\{x\}$, while the strongly packed sets are the singleton $\{y\}$ and the full set $\{x,y\}$.

Theorems & Definitions (73)

• Theorem 1.1: KlyKT99
• Definition 2.1
• Definition 2.2
• Lemma 2.3: Embedding lemma
• Theorem 2.4
• Corollary 2.5
• Remark 2.6
• Theorem 2.7: The self-characterisation theorem
• Remark 2.8
• Remark 2.9