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

Gap Universality of Generalized Wigner and beta-Ensembles