Fixed energy universality for generalized wigner matrices

P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied Mathematics 69 (2016) 1815–1881.


Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2016-10-01
Journal Title
Communications on Pure and Applied Mathematics
Acknowledgement
The work of P.B. was partially supported by National Sci- ence Foundation Grant DMS-1208859. The work of L.E. was partially supported by ERC Advanced Grant RANMAT 338804. The work of H.-T. Y. was partially supported by National Science Foundation Grant DMS-1307444 and a Simons In- vestigator award. The work of J.Y. was partially supported by National Science Foundation Grant DMS-1207961. The major part of this research was conducted when all authors were visiting IAS and were also supported by National Science Foundation Grant DMS-1128255.
Volume
69
Issue
10
Page
1815 - 1881
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Cite this

Bourgade P, Erdös L, Yau H, Yin J. Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. 2016;69(10):1815-1881. doi:10.1002/cpa.21624
Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2016). Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics, 69(10), 1815–1881. https://doi.org/10.1002/cpa.21624
Bourgade, Paul, László Erdös, Horngtzer Yau, and Jun Yin. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics 69, no. 10 (2016): 1815–81. https://doi.org/10.1002/cpa.21624.
P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Fixed energy universality for generalized wigner matrices,” Communications on Pure and Applied Mathematics, vol. 69, no. 10, pp. 1815–1881, 2016.
Bourgade P, Erdös L, Yau H, Yin J. 2016. Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. 69(10), 1815–1881.
Bourgade, Paul, et al. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics, vol. 69, no. 10, Wiley-Blackwell, 2016, pp. 1815–81, doi:10.1002/cpa.21624.

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