Establish soft-edge rigidity for truncated-entry covariance and Wigner matrices

Establish soft-edge rigidity estimates, including the required control of the largest eigenvalue, for Wigner and sample covariance matrices with truncated entries having only four finite moments.

Background

The thesis proves hard-edge rigidity for square sample covariance matrices under four-moment and truncation assumptions. It notes that analogous soft-edge arguments require control of the largest eigenvalue, but that such control is unavailable under the stated truncated-entry assumptions for both Wigner and sample covariance ensembles. Consequently, extending rigidity estimates to the soft edge in this low-moment setting remains unresolved.

References

Here we focus on hard-edge rigidity, since proofs of soft-edge rigidity require control of the largest eigenvalue which, to our knowledge, is not currently available in the case of truncated entries with four moments, in either the Wigner or the Sample Covariance case.

— 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”