Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

Paul Bourgade; Patrick Lopatto; Ofer Zeitouni

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arXiv · 2312.13335

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

Paul Bourgade·Patrick Lopatto·Ofer Zeitouni

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Abstract

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and $β$-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov--Hiary--Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices. Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.

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Paul Bourgade, Patrick Lopatto, Ofer Zeitouni. 2024-08-10. Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices. https://arxiv.org/abs/2312.13335

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