Optimal Rigidity and Maximum of the Characteristic Polynomial of Wigner Matrices
Paul Bourgadea, Patrick Lopattob, Ofer Zeitounic✉
- aCourant Institute, New York University, New York, NY, USA
- bDepartment of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC, USA
- cDepartment of Mathematics, Weizmann Institute of Science, Rehovot, Israel
- ✉Corresponding author
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Abstract
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.