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Gap Universality of Generalized Wigner and beta-Ensembles

We consider generalized Wigner ensembles and general beta-ensembles with analytic potentials for any beta larger than 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian beta-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal.