On the randomized Horn problem and the surface tension of hives

TL;DR

This work analyzes the randomized Horn problem through the lens of hive combinatorics, linking large-deviation behavior of eigenvalue sums to a surface-tension framework. It develops a Gaussian-integral/maximum-entropy formalism for augmented hives, derives limiting volumes for GUE-driven hive polytopes, and provides explicit bounds and a continuum-entropy expression for the surface tension. The paper also presents an exact sampling scheme for augmented hives, leverages Gelfand–Tsetlin patterns and the minor process, and demonstrates numerical experiments with lozenge tilings to illustrate the surface-tension landscape. By connecting random matrix theory, Hive combinatorics, and statistical-physics ideas, it offers quantitative bounds, entropy formulas, and practical algorithms for simulating and estimating surface tension in hive models.

Abstract

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.

Figures (21)

• Figure 1.1: A triangle with $n=5$.
• Figure 1.2: The three types of rhombi.
• Figure 1.3: An augmented hive for $n = 4$. The right triangle below the main diagonal of the $4\times 4$ square corresponds to a random Gelfand-Tsetlin (GT) pattern (printed in red at the center of a diagonal edge) with top row $\nu_n$, given by the difference of the number at a vertex and the one southwest to it. The right triangle above the main diagonal corresponds to a random hive with boundary conditions, $(\lambda_4, \mu_4, \nu_4)$.
• Figure 1.4: Gelfand-Tsetlin pattern for $n=5$.
• Figure 5.1: Minor Process with $n=3$.

Theorems & Definitions (42)

• Definition 1
• Definition 2
• Definition 3
• Definition 4: Discrete hive
• Definition 5: Discrete augmented hive
• Definition 6: $C^k$ hives
• Definition 7
• Definition 8
• Definition 9
• Definition 10