Fixed energy universality for generalized Wigner matrices

Paul Bourgade Cambridge University Laszlo Erdos Institute of Science and Technology Austria Hong-Tzer Yau Harvard University Jun Yin University of Wisconsin, Madison

Probability mathscidoc:1608.28003

Best Paper Award in Applied Mathematics in 2018

Communications on Pure and Applied Mathematics, 69, (10), 1815–1881 , 2015
We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics
Universality, Homogenization, Dyson Brownian motion
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@inproceedings{paul2015fixed,
  title={Fixed energy universality for generalized Wigner matrices},
  author={Paul Bourgade, Laszlo Erdos, Hong-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823100227867233397},
  booktitle={Communications on Pure and Applied Mathematics},
  volume={69},
  number={10},
  pages={1815–1881 },
  year={2015},
}
Paul Bourgade, Laszlo Erdos, Hong-Tzer Yau, and Jun Yin. Fixed energy universality for generalized Wigner matrices. 2015. Vol. 69. In Communications on Pure and Applied Mathematics. pp.1815–1881 . http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823100227867233397.
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