{"doi":"10.1214/24-aop1705","date_published":"2025-01-19T00:00:00Z","publication_identifier":{"issn":["0091-1798"]},"citation":{"chicago":"Ji, Hong Chang, and Jaewhi Park. “Tracy-Widom Limit for Free Sum of Random Matrices.” <i>The Annals of Probability</i>. Institute of Mathematical Statistics, 2025. <a href=\"https://doi.org/10.1214/24-aop1705\">https://doi.org/10.1214/24-aop1705</a>.","apa":"Ji, H. C., &#38; Park, J. (2025). Tracy-Widom limit for free sum of random matrices. <i>The Annals of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/24-aop1705\">https://doi.org/10.1214/24-aop1705</a>","short":"H.C. Ji, J. Park, The Annals of Probability 53 (2025) 239–298.","ista":"Ji HC, Park J. 2025. Tracy-Widom limit for free sum of random matrices. The Annals of Probability. 53(1), 239–298.","ieee":"H. C. Ji and J. Park, “Tracy-Widom limit for free sum of random matrices,” <i>The Annals of Probability</i>, vol. 53, no. 1. Institute of Mathematical Statistics, pp. 239–298, 2025.","mla":"Ji, Hong Chang, and Jaewhi Park. “Tracy-Widom Limit for Free Sum of Random Matrices.” <i>The Annals of Probability</i>, vol. 53, no. 1, Institute of Mathematical Statistics, 2025, pp. 239–98, doi:<a href=\"https://doi.org/10.1214/24-aop1705\">10.1214/24-aop1705</a>.","ama":"Ji HC, Park J. Tracy-Widom limit for free sum of random matrices. <i>The Annals of Probability</i>. 2025;53(1):239-298. doi:<a href=\"https://doi.org/10.1214/24-aop1705\">10.1214/24-aop1705</a>"},"volume":53,"type":"journal_article","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331"}],"_id":"19039","OA_place":"repository","OA_type":"green","language":[{"iso":"eng"}],"day":"19","title":"Tracy-Widom limit for free sum of random matrices","quality_controlled":"1","oa_version":"Preprint","year":"2025","issue":"1","department":[{"_id":"LaEr"}],"page":"239 - 298","date_updated":"2025-04-14T07:57:18Z","external_id":{"arxiv":["2110.05147"]},"corr_author":"1","date_created":"2025-02-17T09:32:16Z","publication":"The Annals of Probability","month":"01","acknowledgement":"The work of H.C. Ji was partially supported by ERC Advanced Grant “RMTBeyond” No. 101020331. The work of J. Park was partially supported by National Research Foundation of Korea under grant number NRF-2019R1A5A1028324. The authors would like to thank Ji Oon Lee for helpful discussions.","abstract":[{"text":"We consider fluctuations of the largest eigenvalues of the random matrix model A + UBU∗ where A and B are N × N deterministic Hermitian (or symmetric) matrices and U is a Haar-distributed unitary (or orthogonal) matrix. We prove that the largest eigenvalue weakly converges to the GUE (or GOE) Tracy–Widom distribution, under mild assumptions on A and B to\r\nguarantee that the density of states of the model decays as square root around\r\nthe upper edge. Our proof is based on the comparison of the Green function\r\nalong the Dyson Brownian motion starting from the matrix A + UBU∗ and\r\nending at time N−1/3+o(1). As a byproduct of our proof, we also prove an\r\noptimal local law for the Dyson Brownian motion up to the constant time\r\nscale.","lang":"eng"}],"ec_funded":1,"article_type":"original","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2110.05147"}],"publisher":"Institute of Mathematical Statistics","scopus_import":"1","status":"public","arxiv":1,"intvolume":"        53","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","publication_status":"published","author":[{"first_name":"Hong Chang","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d","full_name":"Ji, Hong Chang","last_name":"Ji"},{"full_name":"Park, Jaewhi","first_name":"Jaewhi","last_name":"Park"}],"oa":1}