How much can the eigenvalues of a random Hermitian matrix fluctuate?
(2021) Duke Mathematical Journal — Vol. 170, n° 9, p. 2085-2235 (2021)
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- Abstract
- The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices -- the Gaussian Unitary Ensemble being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher-Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.
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Claeys, T., Fahs, B., Lambert, G., & Webb, C. (2021). How much can the eigenvalues of a random Hermitian matrix fluctuate? Duke Mathematical Journal, 170(9), 2085-2235. https://hdl.handle.net/2078.5/268917 (Original work published 2021)