Linear Eigenvalue Statistics at the Cusp

TL;DR

This paper proves universal Gaussian fluctuations for mesoscopic linear eigenvalue statistics near cusp singularities in Wigner-type matrices, covering all regimes from edges to bulk, and introduces a new family of functionals for bias and variance.

Contribution

It provides the first study of linear eigenvalue statistics at cusp-like singularities, establishing universality and describing the transition from edges to bulk.

Findings

Gaussian fluctuations at cusps are universal

Identifies a new family of functionals for bias and variance

Complete description of eigenvalue statistics regimes

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.