article published yes Volodymyr Riabov author 1949f904-edfb-11eb-afb5-e2dfddabb93b LaEr department 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. https://research-explorer.ista.ac.at/download/19598/20916/2025_ProbTheoryRelatFields_Riabov.pdf application/pdfno Springer Nature2025 eng 0178-8051 1432-2064 2307.07432 00146699730000110.1007/s00440-025-01373-w 1931183-1237 https://research-explorer.ista.ac.at/record/20575 Riabov, V. (2025). Linear Eigenvalue statistics at the cusp. <i>Probability Theory and Related Fields</i>. Springer Nature. <a href="https://doi.org/10.1007/s00440-025-01373-w">https://doi.org/10.1007/s00440-025-01373-w</a> Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” <i>Probability Theory and Related Fields</i>. Springer Nature, 2025. <a href="https://doi.org/10.1007/s00440-025-01373-w">https://doi.org/10.1007/s00440-025-01373-w</a>. Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” <i>Probability Theory and Related Fields</i>, vol. 193, Springer Nature, 2025, pp. 1183–237, doi:<a href="https://doi.org/10.1007/s00440-025-01373-w">10.1007/s00440-025-01373-w</a>. V. Riabov, “Linear Eigenvalue statistics at the cusp,” <i>Probability Theory and Related Fields</i>, vol. 193. Springer Nature, pp. 1183–1237, 2025. V. Riabov, Probability Theory and Related Fields 193 (2025) 1183–1237. Riabov V. Linear Eigenvalue statistics at the cusp. <i>Probability Theory and Related Fields</i>. 2025;193:1183-1237. doi:<a href="https://doi.org/10.1007/s00440-025-01373-w">10.1007/s00440-025-01373-w</a> Riabov V. 2025. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. 193, 1183–1237. 195982025-04-20T22:01:28Z2026-04-07T12:32:19Z