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15 June 2021 How much can the eigenvalues of a random Hermitian matrix fluctuate?
T. Claeys, B. Fahs, G. Lambert, C. Webb
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Duke Math. J. 170(9): 2085-2235 (15 June 2021). DOI: 10.1215/00127094-2020-0070


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 (GUE) 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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T. Claeys. B. Fahs. G. Lambert. C. Webb. "How much can the eigenvalues of a random Hermitian matrix fluctuate?." Duke Math. J. 170 (9) 2085 - 2235, 15 June 2021.


Received: 18 June 2019; Revised: 15 June 2020; Published: 15 June 2021
First available in Project Euclid: 18 May 2021

Digital Object Identifier: 10.1215/00127094-2020-0070

Primary: 60B20
Secondary: 35Q15 , 47B35 , 60G15 , 60G57

Keywords: eigenvalue rigidity , Fisher–Hartwig singularities of jump type , Gaussian multiplicative chaos , Gaussian unitary ensemble , Riemann–Hilbert asymptotics

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 9 • 15 June 2021
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