Abstract
Suppose α, β are Lipschitz, strongly concave functions from to and γ is a concave function from to such that , and . For an Hermitian matrix W, let denote the vector in whose coordinates are the eigenvalues of W listed in nonincreasing order. Let , on and , at all points of , where is the left derivative. Let , for , and similarly, and .
Let , be independent random Hermitian matrices from unitarily invariant distributions with spectra , , respectively. We define norm to correspond in a certain way to the sup norm of an antiderivative. We prove that the following limit exists:
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 and , in terms of the surface tension σ mentioned above.
We also prove large deviation principles for random hives with α and β that are , 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
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