Papers › On the randomized Horn problem and the surface tension of hives
On the randomized Horn problem and the surface tension of hives
Aalok Gangopadhyay, Hariharan Narayanan
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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.
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