Hives, as defined by Knutson and Tao, are discrete concave functions on a triangular grid on an equilateral triangle of side n. It is known through the work of Knutson and Tao that the probability distribution of the spectrum of the sum of two independent random matrices with unitarily invariant distributions and given spectra can be expressed in terms of certain marginals of random hives.   We prove the existence of a surface tension function sigma depending on the Hessian, for continuum limits of random hives, and prove a large deviation principle for the large n limit of random hives in terms of sigma. Through the aforementioned connection, we also obtain a large deviation principle for the spectrum of the sum of two random matrices with given spectra.   This is joint work with Scott Sheffield.