article published yes Giorgio Cipolloni author 42198EFA-F248-11E8-B48F-1D18A9856A870000-0002-4901-7992 László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Dominik J Schröder author 408ED176-F248-11E8-B48F-1D18A9856A870000-0002-2904-1856 LaEr department Random matrices beyond Wigner-Dyson-Mehta project 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. https://research-explorer.ista.ac.at/download/12148/12356/2022_ForumMath_Cipolloni.pdf application/pdfno Cambridge University Press2022 eng Computational MathematicsDiscrete Mathematics and CombinatoricsGeometry and TopologyMathematical PhysicsStatistics and ProbabilityAlgebra and Number TheoryTheoretical Computer ScienceAnalysis 2050-5094 00087371920000110.1017/fms.2022.86 10 G. Cipolloni, L. Erdös, and D. J. Schröder, “Rank-uniform local law for Wigner matrices,” <i>Forum of Mathematics, Sigma</i>, vol. 10. Cambridge University Press, 2022. Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). Rank-uniform local law for Wigner matrices. <i>Forum of Mathematics, Sigma</i>. Cambridge University Press. <a href="https://doi.org/10.1017/fms.2022.86">https://doi.org/10.1017/fms.2022.86</a> Cipolloni G, Erdös L, Schröder DJ. Rank-uniform local law for Wigner matrices. <i>Forum of Mathematics, Sigma</i>. 2022;10. doi:<a href="https://doi.org/10.1017/fms.2022.86">10.1017/fms.2022.86</a> Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Rank-Uniform Local Law for Wigner Matrices.” <i>Forum of Mathematics, Sigma</i>. Cambridge University Press, 2022. <a href="https://doi.org/10.1017/fms.2022.86">https://doi.org/10.1017/fms.2022.86</a>. Cipolloni, Giorgio, et al. “Rank-Uniform Local Law for Wigner Matrices.” <i>Forum of Mathematics, Sigma</i>, vol. 10, e96, Cambridge University Press, 2022, doi:<a href="https://doi.org/10.1017/fms.2022.86">10.1017/fms.2022.86</a>. Cipolloni G, Erdös L, Schröder DJ. 2022. Rank-uniform local law for Wigner matrices. Forum of Mathematics, Sigma. 10, e96. G. Cipolloni, L. Erdös, D.J. Schröder, Forum of Mathematics, Sigma 10 (2022). 121482023-01-12T12:07:30Z2025-04-14T07:57:18Z