Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

TL;DR

This paper determines the leading order maximum of the characteristic polynomial for Wigner matrices and $eta$-ensembles, establishing universality and optimal rigidity results through advanced probabilistic and dynamical techniques.

Contribution

It provides the first universal results on the Fyodorov--Hiary--Keating conjectures for these models and answers the question of optimal spectral rigidity for Wigner matrices.

Findings

Maximum of characteristic polynomial determined to leading order

Universality of maximum established for Gaussian-divisible Wigner matrices

Proofs combine dynamical eigenvalue techniques with Gaussian multiplicative chaos

Abstract

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and $\beta$-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.