{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"        33","scopus_import":"1","publication_status":"published","page":"2981-3009","status":"public","date_updated":"2024-01-09T08:16:41Z","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"article_processing_charge":"No","date_published":"2023-08-01T00:00:00Z","title":"Local laws for multiplication of random matrices","quality_controlled":"1","acknowledgement":"The first author is partially supported by NSF Grant DMS-2113489 and grateful for the AMS-SIMONS travel grant (2020–2023). The second author is supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nThe authors would like to thank the Editor, Associate Editor and an anonymous referee for their many critical suggestions which have significantly improved the paper. We also want to thank Zhigang Bao and Ji Oon Lee for many helpful discussions and comments.","volume":33,"type":"journal_article","_id":"14750","doi":"10.1214/22-aap1882","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2010.16083"}],"date_created":"2024-01-08T13:03:18Z","oa":1,"publisher":"Institute of Mathematical Statistics","external_id":{"arxiv":["2010.16083"]},"article_type":"original","day":"01","year":"2023","author":[{"first_name":"Xiucai","full_name":"Ding, Xiucai","last_name":"Ding"},{"full_name":"Ji, Hong Chang","last_name":"Ji","first_name":"Hong Chang","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d"}],"language":[{"iso":"eng"}],"issue":"4","department":[{"_id":"LaEr"}],"project":[{"call_identifier":"H2020","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"month":"08","ec_funded":1,"publication_identifier":{"issn":["1050-5164"]},"citation":{"ista":"Ding X, Ji HC. 2023. Local laws for multiplication of random matrices. The Annals of Applied Probability. 33(4), 2981–3009.","ama":"Ding X, Ji HC. Local laws for multiplication of random matrices. <i>The Annals of Applied Probability</i>. 2023;33(4):2981-3009. doi:<a href=\"https://doi.org/10.1214/22-aap1882\">10.1214/22-aap1882</a>","short":"X. Ding, H.C. Ji, The Annals of Applied Probability 33 (2023) 2981–3009.","chicago":"Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.” <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2023. <a href=\"https://doi.org/10.1214/22-aap1882\">https://doi.org/10.1214/22-aap1882</a>.","apa":"Ding, X., &#38; Ji, H. C. (2023). Local laws for multiplication of random matrices. <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/22-aap1882\">https://doi.org/10.1214/22-aap1882</a>","ieee":"X. Ding and H. C. Ji, “Local laws for multiplication of random matrices,” <i>The Annals of Applied Probability</i>, vol. 33, no. 4. Institute of Mathematical Statistics, pp. 2981–3009, 2023.","mla":"Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.” <i>The Annals of Applied Probability</i>, vol. 33, no. 4, Institute of Mathematical Statistics, 2023, pp. 2981–3009, doi:<a href=\"https://doi.org/10.1214/22-aap1882\">10.1214/22-aap1882</a>."},"publication":"The Annals of Applied Probability","oa_version":"Preprint","abstract":[{"lang":"eng","text":"Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N × N deterministic matrices and U is either an N × N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201–220), the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution\r\nof the limiting ESDs of A and B, denoted as μα \u0002 μβ, where μα and μβ are the limiting ESDs of A and B, respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of μA \u0002μB, where μA and μB are the ESDs of A and B, respectively and the associated subordination functions\r\nhave a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of A, B and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model A+UBU∗ in (J. Funct. Anal. 271 (2016) 672–719; Comm. Math.\r\nPhys. 349 (2017) 947–990; Adv. Math. 319 (2017) 251–291; J. Funct. Anal. 279 (2020) 108639), and our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020) 108639) for the multiplicative model. "}]}