Geodesic
Rank-uniform local law for Wigner matrices
Forum of Mathematics, Sigma, Tome 10 (2022)

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We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables. 
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     author = {Giorgio Cipolloni and L\'aszl\'o Erd\H{o}s and Dominik Schr\"oder},
     title = {Rank-uniform local law for {Wigner} matrices},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2022},
     doi = {10.1017/fms.2022.86},
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Giorgio Cipolloni; László Erdős; Dominik Schröder. Rank-uniform local law for Wigner matrices. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.86

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