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Fixed energy universality for generalized Wigner matrices. (English) Zbl 1354.15025
The main result is to prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for real symmetric and Hermitian Wigner matrices. The general assumptions on the entries of the matrices are relatively low: mainly i.i.d. conditions with a typical variance for each entry of order \(N^{-1}\). Though fixed energy universality theorems already existed for Hermitian matrices in the literature, the real symmetric case is proved to be more challenging and it is the main contribution of this article to propose a proof in this setting. Apart from the result itself, the second great interest of the article is to provide a new approach on the problem with a new strategy to prove the result involving the Dyson Brownian motion. In particular, this method could be used in other contexts and may interest many experts in the field. For their purposes, the authors efficiently use a combination of tools arising in probability, analysis and stochastic differential equations that require a solid background on those fields. Thus, the article may appear technical but since each step is properly detailed, it still remains accessible and pleasant to read.

MSC:
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60J65 Brownian motion
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