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<p>We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and <i>β</i>-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.</p><p>Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.</p>

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Optimal Rigidity and Maximum of the Characteristic Polynomial of Wigner Matrices

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年2月5日

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  • Paul Bourgade,
  • Patrick Lopatto,
  • Ofer Zeitouni

摘要

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.