Linear Eigenvalue Statistics at the cusp

Area of Science:

• Mathematics
• Physics
• Statistical Mechanics

Background:

• Wigner-type random matrices are fundamental in statistical mechanics and quantum chaos.
• Spectral density singularities, particularly cusps, pose challenges for understanding eigenvalue statistics.
• Previous studies lacked analysis of linear eigenvalue statistics at cusp-like singularities.

Purpose of the Study:

• To establish universal Gaussian fluctuations for mesoscopic linear eigenvalue statistics near cusp-like singularities.
• To analyze the transitionary regime from regular edges to sharp cusps and the bulk.
• To provide a complete description of eigenvalue statistics in all possible regimes.

Main Methods:

• Analysis of mesoscopic linear eigenvalue statistics.
• Investigation of Wigner-type random matrices.
• Development of new functionals for bias and variance.

Main Results:

• Universal Gaussian fluctuations are established for eigenvalue statistics near cusps.
• A new one-parameter family of functionals governing bias and variance is identified.
• The analysis covers the entire transitionary regime, from regular edges to sharp cusps and the bulk.

Conclusions:

• This work provides a complete description of linear eigenvalue statistics in all regimes of Wigner-type random matrices.
• The identified functionals interpolate between known formulas for bulk and edge statistics.
• The findings offer new insights into the behavior of random matrices near spectral singularities.