Mesoscopic eigenvalue statistics for Wigner-type matrices

Riabov V. 2025. Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. 61(1), 129–154.

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Abstract
We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions. The main novel ingredient is an optimal local law for the two-point function $T(z,\zeta)$ and a general class of related quantities involving two resolvents at nearby spectral parameters.

On établit un théorème limite central universel pour les statistiques linéaires mésoscopiques des valeurs propres d’une matrice de type Wigner au milieu du spectre, avec des fonctions de classe et à support compact. La principale nouveauté de cette approche est qu’elle repose sur une loi locale optimale pour la fonction à deux points $T(z,\zeta)$ , ainsi que pour une classe plus générale d’observables impliquant deux résolvantes évaluées en des paramètres proches.
Publishing Year
Date Published
2025-02-01
Journal Title
Annales de l'institut Henri Poincare (B) Probability and Statistics
Publisher
Institute of Mathematical Statistics
Acknowledgement
I would like to express my gratitude to László Erdős for suggesting the project and supervising my work. I am also thankful to Yuanyuan Xu and Oleksii Kolupaiev for many helpful discussions. Furthermore, I am grateful to Guillaume Dubach for translating the abstract into French. The author was supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.
Volume
61
Issue
1
Page
129-154
ISSN
IST-REx-ID

Cite this

Riabov V. Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. 2025;61(1):129-154. doi:10.1214/23-AIHP1438
Riabov, V. (2025). Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/23-AIHP1438
Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.” Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics, 2025. https://doi.org/10.1214/23-AIHP1438.
V. Riabov, “Mesoscopic eigenvalue statistics for Wigner-type matrices,” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 61, no. 1. Institute of Mathematical Statistics, pp. 129–154, 2025.
Riabov V. 2025. Mesoscopic eigenvalue statistics for Wigner-type matrices. Annales de l’institut Henri Poincare (B) Probability and Statistics. 61(1), 129–154.
Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 61, no. 1, Institute of Mathematical Statistics, 2025, pp. 129–54, doi:10.1214/23-AIHP1438.
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