Anisotropic local law near the hard edge

Establish the anisotropic local law for sample covariance matrices with independently sampled, centered columns satisfying quadratic-form concentration, polynomial norm moments, and the stated covariance nondegeneracy condition on regular spectral domains approaching the origin, in particular near the hard edge when the aspect ratio n/N tends to one.

Background

The paper proves an optimal anisotropic local law for sample covariance matrices under quadratic-form concentration, but its regular spectral domains are required to remain bounded away from the origin. This restriction arises from the square-root reparametrization used in the block linearization and characteristic-flow argument.

The unresolved extension concerns spectral domains approaching zero, including the hard-edge regime when n/N converges to one. The authors note that analogous results are known for independent-entry models, but not under the general within-column dependence allowed by their quadratic-form concentration framework.

References

First, our regular spectral domains are bounded away from the origin, as required by the square-root reparametrization of Section \ref{sec:block}. Establishing the anisotropic law on regular domains approaching zero, in particular near the hard edge when $n/N \to 1$, remains open; for independent entries, see .

— The anisotropic local law for sample covariance matrices under quadratic-form concentration  (2609.09440 - Ma et al., 8 Sep 2026) in Section 2, subsection “Discussion and examples,” paragraph headed “Open problems.”