On the randomized Horn problem and the surface tension of hives

TL;DR

This paper investigates the asymptotic behavior of the randomized Horn problem, providing bounds on the surface tension of hives, a closed-form entropy expression, and empirical results related to large random hives and lozenge tilings.

Contribution

It introduces bounds on the surface tension function, derives a closed-form entropy formula for continuum hives, and presents empirical analyses of large random hives and tilings.

Findings

Bounds on the surface tension function are established.

A closed-form expression for the total entropy of minimal surface tension hives is derived.

Empirical results demonstrate properties of random hives and lozenge tilings for large sizes.

Abstract

Given two nonincreasing $n$-tuples of real numbers $\lambda_n$, $\mu_n$, the Horn problem asks for a description of all nonincreasing $n$-tuples of real numbers $\nu_n$ 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.