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A necessary and sufficient condition for edge universality of Wigner matrices. (English) Zbl 1296.60007

It is by now classical that for the largest eigenvalue \(\lambda_N\) of a Gaussian (Orthogonal, Unitary, or Symplectic) Ensemble, the statistic \(N^{2/3}(\lambda_N - 2)\) converges weakly to the corresponding Tracy-Widom law. In the paper under review, the authors provide a remarkably simple criterion for this to remain true for a Wigner matrix with i.i.d.centered off-diagional entries of variance \(1\) (and i.i.d.centered diagonal entries of finite variance). In fact, this is true if, and only if, \(\lim_{s \to +\infty}\;s^4\;\mathbb{P}\left( |x_{12}|\geq s\right) = 0.\)

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
81V70 Many-body theory; quantum Hall effect
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
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