Determine hard-edge eigenvalue fluctuation laws

Determine fluctuation estimates for individual eigenvalues at the hard edge of the sample covariance spectrum, comparable to the known bulk and soft-edge fluctuation results for Gaussian unitary and Wishart ensembles.

Background

The thesis obtains hard-edge rigidity bounds of order approximately logarithmic corrections times N{-2} for the individual eigenvalues of square sample covariance matrices. It contrasts these bounds with sharper fluctuation results known in the bulk and at the soft edge for Gaussian unitary and Wishart ensembles. The corresponding hard-edge fluctuation theory is explicitly identified as unavailable.

References

To our knowledge similar results are not yet available for the "hard" edge.

— Spectral properties of Random Matrices  (2609.11011 - Kafetzopoulos, 10 Sep 2026) in Chapter 1, Section “Discussion about the new results,” subsection “1. Local Marchenko-Pastur law”