Universality for Wigner Matrices: Optimal Moment Conditions
§ Problem Statement
§ Discussion
§ Significance & Implications
§ Known Partial Results
§ References
Universality of local spectral statistics of random matrices
László Erdős, Horng-Tzer Yau (2012)
Bulletin of the AMS
📍 Section 11 (Historical Remarks), edge-universality discussion (“the finiteness of the fourth moment was conjectured to be sufficient” and open question on removing the $12+\varepsilon$ moment condition), p. 408 (Bull. Amer. Math. Soc. 49 (2012), 377–414).
Random matrices: Universality of local eigenvalue statistics
Terence Tao, Van Vu (2011)
Acta Mathematica
On the distribution of the roots of certain symmetric matrices
Eugene P. Wigner (1958)
Annals of Mathematics
📍 Main theorem (semicircle law for eigenvalue distribution of real symmetric random matrices), pp. 325–327.
Level-spacing distributions and the Airy kernel
Craig A. Tracy, Harold Widom (1994)
Communications in Mathematical Physics
📍 Section 1 (Introduction), Theorem 1 (Tracy–Widom distribution F₂ for GUE largest eigenvalue fluctuations), p. 152.