Edge universality under quadratic-form concentration

Determine whether the basic covariance assumptions and quadratic-form concentration assumption alone imply edge universality at every uniformly regular soft edge bounded away from zero for sample covariance matrices with possibly dependent column coordinates.

Background

At uniformly regular soft edges bounded away from zero, the paper combines prior optimal eigenvalue rigidity with its new anisotropic local law under the basic covariance assumptions and quadratic-form concentration. These results establish strong local spectral control but do not establish the limiting distribution of the edge eigenvalues.

The open question is whether those two assumptions suffice for edge universality at every regular soft edge. The paper contrasts this unresolved setting with broad independent-entry results and with the Tracy–Widom theorem for isotropic log-concave columns under an additional unconditionality assumption.

References

It is therefore natural to ask whether these two assumptions alone imply edge universality at every regular soft edge.

— 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.”