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

Abstract

Suppose α, β are Lipschitz, strongly concave functions from $[0,1]$ to $\mathbb{R}$ and γ is a concave function from $[0,1]$ to $\mathbb{R}$ such that $\mathit{\alpha }(0)=\mathit{\gamma }(0)=0$, $\mathit{\alpha }(1)=\mathit{\beta }(0)=0$ and $\mathit{\beta }(1)=\mathit{\gamma }(1)=0$. For an $\mathit{n}\times \mathit{n}$ Hermitian matrix W, let $\mathrm{spec}(\mathit{W})$ denote the vector in ${\mathbb{R}}^{\mathit{n}}$ whose coordinates are the eigenvalues of W listed in nonincreasing order. Let $\mathit{\lambda }={\partial }^{-}\mathit{\alpha }$, $\mathit{\mu }={\partial }^{-}\mathit{\beta }$ on $(0,1]$ and $\mathit{\nu }={\partial }^{-}\mathit{\gamma }$, at all points of $(0,1]$, where ${\partial }^{-}$ is the left derivative. Let ${\mathit{\lambda }}_{\mathit{n}}(\mathit{i}):={\mathit{n}}^2(\mathit{\alpha }(\frac{\mathit{i}}{\mathit{n}})-\mathit{\alpha }(\frac{\mathit{i}-1}{\mathit{n}}))$, for $\mathit{i}\in [\mathit{n}]$, and similarly, ${\mathit{\mu }}_{\mathit{n}}(\mathit{i}):={\mathit{n}}^2(\mathit{\beta }(\frac{\mathit{i}}{\mathit{n}})-\mathit{\beta }(\frac{\mathit{i}-1}{\mathit{n}}))$ and ${\mathit{\nu }}_{\mathit{n}}(\mathit{i}):={\mathit{n}}^2(\mathit{\gamma }(\frac{\mathit{i}}{\mathit{n}})-\mathit{\gamma }(\frac{\mathit{i}-1}{\mathit{n}}))$.

Let ${\mathit{X}}_{\mathit{n}}$, ${\mathit{Y}}_{\mathit{n}}$ be independent random Hermitian matrices from unitarily invariant distributions with spectra ${\mathit{\lambda }}_{\mathit{n}}$, ${\mathit{\mu }}_{\mathit{n}}$, respectively. We define norm $\Vert ·{\Vert }_{\mathit{I}}$ to correspond in a certain way to the sup norm of an antiderivative. We prove that the following limit exists:

$$\underset{\mathit{n}\to \infty }{lim}\frac{log\mathbb{P}[\Vert \mathrm{spec}({\mathit{X}}_{\mathit{n}}\mathbf{+}{\mathit{Y}}_{\mathit{n}})-{\mathit{\nu }}_{\mathit{n}}{\Vert }_{\mathit{I}}<{\mathit{n}}^2\mathit{ϵ}]}{{\mathit{n}}^2}.$$

We interpret this limit in terms of the surface tension σ of continuum limits of the discrete hives defined by Knutson and Tao.

We provide matching large deviation upper and lower bounds for the spectrum of the sum of two random matrices ${\mathit{X}}_{\mathit{n}}$ and ${\mathit{Y}}_{\mathit{n}}$, in terms of the surface tension σ mentioned above.

We also prove large deviation principles for random hives with α and β that are ${\mathit{C}}^2$, where the rate function can be interpreted in terms of the maximizer of a functional that is the sum of a term related to the free energy of hives associated with α, β and γ and a quantity related to logarithms of Vandermonde determinants associated with γ.