Large deviations for random hives and the spectrum of the sum of two random matrices

Suppose $\alpha, \beta$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $\gamma$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $\alpha(0) = \gamma(0) = 0$, and $\alpha(1) = \beta(0) = 0$ and $\beta(1) = \gamma(1) = 0.$ For an $n \times n$ Hermitian matrix $W$, let $spec(W)$ denote the vector in $\mathbb{R}^n$ whose coordinates are the eigenvalues of $W$ listed in non-increasing order. Let $\lambda = \partial^- \alpha$, $\mu = \partial^- \beta$ on $(0, 1]$ and $\nu = \partial^- \gamma,$ at all points of $(0, 1]$, where $\partial^-$ is the left derivative, which is monotonically decreasing. Let $\lambda_n(i) := n^2(\alpha(\frac{i}{n})-\alpha(\frac{i-1}{n}))$, for $i \in [n]$, and similarly, $\mu_n(i) := n^2(\beta(\frac{i}{n})-\beta(\frac{i-1}{n}))$, and $\nu_n(i) := n^2(\gamma(\frac{i}{n})-\gamma(\frac{i-1}{n}))$. Let $X_n, Y_n$ be independent random Hermitian matrices from unitarily invariant distributions with spectra $\lambda_n$, $\mu_n$ respectively. We define norm $\|\cdot\|_\mathcal{I}$ to correspond in a certain way to the sup norm of an antiderivative. For suitable $\lambda$ and $\mu$, we prove that the following limit exists. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - \nu_n\|_{\mathcal{I}} < n^2 \epsilon\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension $\sigma$ of continuum limits of the discrete hives defined by Knutson and Tao.