Deterministic subexponential approximation of the Horn density function

Determine whether the Horn density function at a prescribed spectrum can be deterministically approximated within a multiplicative factor of exp(o(n^2)).

Background

The Horn density function describes the probability density of the spectrum of the sum of two independent random Hermitian matrices drawn from unitarily invariant distributions with prescribed spectra. The paper notes that this density can be expressed in terms of volumes of hive polytopes, whose dimensions are asymptotically n2/2.

The question is presented as a related open problem extending the deterministic approximation perspective beyond Kostka-polytope volumes. Resolving it would yield a deterministic subexponential-factor approximation for a density arising in the additive random-matrix and Horn-problem setting.

References

Another related open question is to deterministically approximate, within a multiplicative factor of \exp(o(n2)), the Horn density function at a given spectrum in n.

— Deterministically approximating the volume of a Kostka polytope  (2503.06459 - Narayanan et al., 9 Mar 2025) in Section 1, subsection “Discussion”