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Gap universality of generalized Wigner and β -ensembles

László Erdős;  Horng-Tzer Yau — 2015

Journal of the European Mathematical Society

We consider generalized Wigner ensembles and general β -ensembles with analytic potentials for any β ≥ 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 β -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. In fact,...

The local relaxation flow approach to universality of the local statistics for random matrices

László Erdős;  Benjamin Schlein;  Horng-Tzer Yau;  Jun Yin — 2012

Annales de l'I.H.P. Probabilités et statistiques

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues { }=1 are close to their classical location { }=1 determined by the limiting density...

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