Linear Eigenvalue statistics at the cusp

Riabov V. 2025. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields.

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Abstract
We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics in the vicinity of the cusp-like singularities of the limiting spectral density for Wigner-type random matrices. Prior to this work, the linear eigenvalue statistics at the cusp-like singularities were not studied in any ensemble. Our analysis covers not only the exact cusps but the entire transitionary regime from the square-root singularity at a regular edge through the sharp cusp to the bulk. We identify a new one-parameter family of functionals that govern the limiting bias and variance, continuously interpolating between the previously known formulas in the bulk and at a regular edge. Since cusps are the only possible singularities besides the regular edges, our result gives a complete description of the linear eigenvalue statistics in all regimes.
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Date Published
2025-04-15
Journal Title
Probability Theory and Related Fields
Publisher
Springer Nature
Acknowledgement
I would like to express my gratitude to László Erdős for his careful guidance and supervision of my work. I am also thankful to Jana Reker and Joscha Henheik for many helpful discussions. Open access funding provided by Institute of Science and Technology (IST Austria).
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Riabov V. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. 2025. doi:10.1007/s00440-025-01373-w
Riabov, V. (2025). Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-025-01373-w
Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” Probability Theory and Related Fields. Springer Nature, 2025. https://doi.org/10.1007/s00440-025-01373-w.
V. Riabov, “Linear Eigenvalue statistics at the cusp,” Probability Theory and Related Fields. Springer Nature, 2025.
Riabov V. 2025. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields.
Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” Probability Theory and Related Fields, Springer Nature, 2025, doi:10.1007/s00440-025-01373-w.
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