On the randomised Horn problem and the surface tension of hives

Abstract

Given two nonincreasing n-tuples of real numbers l_n, m_n, the Horn problem asks for a description of all nonincreasing n-tuples of real numbers u_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 tends to infinity, large deviations for the randomized Horn problem were given in joint work with Sheffield in terms of a notion of surface tension for hives. In this talk, we discuss 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. This is a joint work with Aalok Gangopadhyay.