How much can the eigenvalues of a random Hermitian matrix fluctuate?

T. Claeys, B. Fahs, G. Lambert, C. Webb

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)

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 (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.

Original languageEnglish
Pages (from-to)2085-2235
JournalDuke Mathematical Journal
Volume170
Issue number9
DOIs
Publication statusPublished - 15 Jun 2021
MoE publication typeA1 Journal article-refereed

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