Universality beyond GOE and behavior for heavy-tailed Wigner matrices

Identify the weakest moment and entry-distribution conditions under which the low-degree polynomial approximation results for spiked GOE and GOE remain valid, and determine how the results must be modified when those conditions fail, particularly for heavy-tailed Wigner matrices.

Background

The paper proves its results for exactly Gaussian GOE noise and presents numerical evidence suggesting universality for Rademacher noise. It conjectures that sufficiently strong moment assumptions should extend the spiked-model and pure-GOE results to more general Wigner matrices, while warning that heavy-tailed matrices may violate the eigenvalue-rigidity inputs used in the proofs. The precise threshold conditions for validity and the appropriate replacement statements outside that regime are left unresolved.

References

Identifying the conditions under which our results continue to hold and how the results need to be modified when these conditions fail constitute an interesting future direction.

— Eigenvalue and Eigenvector Approximation for Random Matrices Using Low-Degree Polynomials  (2609.28781 - Zhang, 23 Sep 2026) in Section 7, Discussion and future directions, paragraph “Universality and non-universality”