Cusp universality for correlated random matrices

preprint Cusp universality for correlated random matrices draft László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Sven Joscha Henheik author 31d731d7-d235-11ea-ad11-b50331c8d7fb0000-0003-1106-327X Volodymyr Riabov author 1949f904-edfb-11eb-afb5-e2dfddabb93b LaEr department Random matrices beyond Wigner-Dyson-Mehta project For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality [arXiv:1804.07744], while cusp universality was proven only for Wigner-type matrices with independent entries [arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus also re-establishing universality of the local eigenvalue statistics in the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744] regimes. 2024 eng arXiv 2410.0681310.48550/arXiv.2410.06813 https://research-explorer.ista.ac.at/record/20322 https://research-explorer.ista.ac.at/record/20575 https://research-explorer.ista.ac.at/record/19540 Erdös, L., Henheik, S. J., & Riabov, V. (n.d.). Cusp universality for correlated random matrices. arXiv. <a href="https://doi.org/10.48550/arXiv.2410.06813">https://doi.org/10.48550/arXiv.2410.06813</a> Erdös, László, et al. “Cusp Universality for Correlated Random Matrices.” ArXiv, doi:<a href="https://doi.org/10.48550/arXiv.2410.06813">10.48550/arXiv.2410.06813</a>. Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices. arXiv, <a href="https://doi.org/10.48550/arXiv.2410.06813">10.48550/arXiv.2410.06813</a>. Erdös, László, Sven Joscha Henheik, and Volodymyr Riabov. “Cusp Universality for Correlated Random Matrices.” ArXiv, n.d. <a href="https://doi.org/10.48550/arXiv.2410.06813">https://doi.org/10.48550/arXiv.2410.06813</a>. L. Erdös, S. J. Henheik, and V. Riabov, “Cusp universality for correlated random matrices,” arXiv. . Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices. arXiv. doi:<a href="https://doi.org/10.48550/arXiv.2410.06813">10.48550/arXiv.2410.06813</a> L. Erdös, S.J. Henheik, V. Riabov, ArXiv (n.d.). 195472025-04-11T08:48:21Z2026-04-07T12:37:11Z