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Title: Quantum unique ergodicity for generalized Wigner matrices.

Abstract: We prove a strong form of delocalization of eigenvectors for general large random matrices called Quantum Unique Ergodicity. This property was first given as a conjecture by Rudnick and Sarnak on the eigenfunctions of the Laplacian on negatively curved compact Riemannian manifolds. In the context of random matrix theory, these estimates state that the mass of an eigenvector over a subset of entries tends to the uniform distribution with very high probability. We are also able to prove that the fluctuations around the uniform distribution are Gaussian for a regime of subsets of entries. The proof relies on new eigenvector observables studied dynamically through the Dyson Brownian motion combined with a novel bootstrap comparison argument. If time allows, after describing the sketch of the dynamical method in random matrix theory, we will develop one of these arguments.

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