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Edge Universality of Beta Ensembles

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Abstract

We prove the edge universality of the beta ensembles for any \({\beta \ge 1}\), provided that the limiting spectrum is supported on a single interval, and the external potential is \({\fancyscript{C}^4}\) and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class \({\fancyscript{C}^4}\).

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Paul Bourgade & Horng-Tzer Yau

  2. Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    László Erdös

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  1. Paul Bourgade
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  2. László Erdös
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  3. Horng-Tzer Yau
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Correspondence to Horng-Tzer Yau.

Additional information

Communicated by H. Spohn

P. Bourgade was partially supported by NSF grant DMS-1208859.

L. Erdös was partially supported by SFB-TR 12 Grant of the German Research Council. On leave from Institute of Mathematics, University of Munich, Germany.

H.-T. Yau was partially supported by NSF grant DMS1307444 and Simons Investigator Award.

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Bourgade, P., Erdös, L. & Yau, HT. Edge Universality of Beta Ensembles. Commun. Math. Phys. 332, 261–353 (2014). https://doi.org/10.1007/s00220-014-2120-z

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