[{"publication":"arXiv","date_created":"2025-04-11T08:48:21Z","title":"Cusp universality for correlated random matrices","month":"11","doi":"10.48550/arXiv.2410.06813","date_updated":"2026-04-07T12:37:11Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","abstract":[{"text":"For correlated real symmetric or complex Hermitian random matrices, we prove\r\nthat the local eigenvalue statistics at any cusp singularity are universal.\r\nSince the density of states typically exhibits only square root edge or cubic\r\nroot cusp singularities, our result completes the proof of the\r\nWigner-Dyson-Mehta universality conjecture in all spectral regimes for a very\r\ngeneral class of random matrices. Previously only the bulk and the edge\r\nuniversality were established in this generality [arXiv:1804.07744], while cusp\r\nuniversality was proven only for Wigner-type matrices with independent entries\r\n[arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an\r\noptimal local law at the cusp using the Zigzag strategy, a recursive tandem of\r\nthe characteristic flow method and a Green function comparison argument.\r\nMoreover, our proof of the optimal local law holds uniformly in the spectrum,\r\nthus also re-establishing universality of the local eigenvalue statistics in\r\nthe previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744]\r\nregimes.","lang":"eng"}],"day":"03","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2410.06813"}],"project":[{"grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020"}],"date_published":"2024-11-03T00:00:00Z","oa":1,"year":"2024","arxiv":1,"status":"public","type":"preprint","related_material":{"record":[{"relation":"later_version","id":"20322","status":"public"},{"relation":"dissertation_contains","id":"20575","status":"public"},{"relation":"dissertation_contains","status":"public","id":"19540"}]},"language":[{"iso":"eng"}],"author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"first_name":"Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","full_name":"Henheik, Sven Joscha","last_name":"Henheik","orcid":"0000-0003-1106-327X"},{"id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","first_name":"Volodymyr","last_name":"Riabov","full_name":"Riabov, Volodymyr"}],"department":[{"_id":"LaEr"}],"_id":"19547","article_processing_charge":"No","OA_place":"repository","acknowledgement":"Supported by the ERC Advanced Grant \"RMTBeyond\"\r\nNo. 101020331.","oa_version":"Preprint","external_id":{"arxiv":["2410.06813"]},"publication_status":"draft","ec_funded":1,"corr_author":"1","citation":{"ieee":"L. Erdös, S. J. Henheik, and V. Riabov, “Cusp universality for correlated random matrices,” <i>arXiv</i>. .","chicago":"Erdös, László, Sven Joscha Henheik, and Volodymyr Riabov. “Cusp Universality for Correlated Random Matrices.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2410.06813\">https://doi.org/10.48550/arXiv.2410.06813</a>.","ista":"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>.","ama":"Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2410.06813\">10.48550/arXiv.2410.06813</a>","short":"L. Erdös, S.J. Henheik, V. Riabov, ArXiv (n.d.).","apa":"Erdös, L., Henheik, S. J., &#38; Riabov, V. (n.d.). Cusp universality for correlated random matrices. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2410.06813\">https://doi.org/10.48550/arXiv.2410.06813</a>","mla":"Erdös, László, et al. “Cusp Universality for Correlated Random Matrices.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/arXiv.2410.06813\">10.48550/arXiv.2410.06813</a>."}}]