On the randomized Horn problem and the surface tension of hives

Given two nonincreasing $n$-tuples of real numbers $λ_n$, $μ_n$, the Horn problem asks for a description of all nonincreasing $n$-tuples of real numbers $ν_n$ such that there exist Hermitian matrices $X_n$, $Y_n$ and $Z_n$ respectively with these spectra such that $X_n + Y_n = Z_n$. There is also a randomized version of this problem where $X_n$ and $Y_n$ are sampled uniformly at random from orbits of Hermitian matrices arising from the conjugacy action by elements of the unitary group. One then asks for a description of the probability measure of the spectrum of the sum $Z_n$. Both the original Horn problem and its randomized version have solutions using the hives introduced by Knutson and Tao. In an asymptotic sense, as $n \rightarrow \infty$, large deviations for the randomized Horn problem were given by Narayanan and Sheffield in terms of the surface tension of hives. In this paper, we provide upper and lower bounds on this surface tension function. We also obtain a closed-form expression for the total entropy of a surface tension minimizing continuum hive with boundary conditions arising from GUE eigenspectra. Finally, we give several empirical results for random hives and lozenge tilings arising from an application of the octahedron recurrence for large $n$ and a numerical approximation of the surface tension function.

💡 Research Summary

The paper investigates the randomized Horn problem, which asks for the distribution of eigenvalues of the sum Zₙ = Xₙ + Yₙ when Xₙ and Yₙ are independent Hermitian matrices drawn uniformly from the unitary orbits determined by prescribed spectra λₙ and μₙ. Building on the classical Horn problem and the hive model of Knutson and Tao, the authors connect the probability density of spec Zₙ to the product of Vandermonde determinants V(·) and the volume of a hive polytope |Hₙ(λₙ, μₙ; νₙ)| (Coquereaux‑Zuber theorem).

A large‑deviation principle due to Narayanan and Sheffield shows that, as n→∞, the logarithmic probability of the spectrum lying in a small ℓ‑∞ ball around νₙ scales like –(n²/2) I(ν), where the rate functional I(γ′) consists of a term ln V(λ) + ln V(μ) + ln V(ν′) and a variational integral involving a convex surface‑tension function σ defined on ℝ₊³. The functional I is minimized over continuous hives h∈H(λ, μ; ν′), and σ appears as the integrand applied to the absolutely continuous part of the inverse Hessian of h.

The core contributions are explicit upper and lower bounds for σ. The authors first develop a Gaussian integral identity (Theorem 6) and handle negative variance parameters (Lemma 1). Lemma 2 and Lemma 3 provide a maximum‑entropy (Boltzmann) principle that identifies the limiting Vandermonde functional V(·). Using these tools they obtain log‑volume bounds for hive polytopes (Theorems 8–10).

The main lower bound (Theorem 3) states that for any admissible boundary functions λ, μ, ν with L²‑norms σ_λ, σ_μ, σ_ν forming a non‑degenerate Euclidean triangle of area Δ>0, \