2024/10/16 by Gangopadhyay, Aalok, Hariharan Narayanan, Narayanan, Hariharan
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2410.12619
openalex publication_date 2024/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given two nonincreasing n-tuples of real numbers λn, μn, the Horn problem asks for a description of all nonincreasing n-tuples of real numbers νn such that there exist Hermitian matrices Xn, Yn and Zn respectively with these spectra such that Xn + Yn = Zn. There is also a randomized version of this problem where Xn and Yn 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 Zn. 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 → ∞, 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.