May 2024 Large deviations for random hives and the spectrum of the sum of two random matrices
Hariharan Narayanan, Scott Sheffield
Ann. Probab. 52(3): 1093-1152 (May 2024). DOI: 10.1214/24-AOP1687

## Abstract

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

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 $‖·{‖}_{\mathit{I}}$ to correspond in a certain way to the sup norm of an antiderivative. We prove that the following limit exists:

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

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.

## Funding Statement

Hariharan Narayanan is partially supported by a Ramanujan fellowship and a Swarna Jayanti fellowship, instituted by the Government of India.
Scott Sheffield is partially supported by NSF awards DMS-1712862 and DMS-2153742.

## Acknowledgments

We are very grateful to the anonymous reviewer for an exceptionally painstaking and careful review that pointed out several inaccuracies. We thank Terence Tao for his valuable comments.

## Citation

Hariharan Narayanan. Scott Sheffield. "Large deviations for random hives and the spectrum of the sum of two random matrices." Ann. Probab. 52 (3) 1093 - 1152, May 2024. https://doi.org/10.1214/24-AOP1687

## Information

Received: 1 January 2022; Revised: 1 February 2024; Published: May 2024
First available in Project Euclid: 23 April 2024

Digital Object Identifier: 10.1214/24-AOP1687

Subjects:
Primary: 60B20 , 60F10
Secondary: 82B41

Keywords: large deviations , random matrices , Random surfaces