--- _id: '14775' abstract: - lang: eng text: We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble. acknowledgement: K. Schnelli was supported by the Swedish Research Council Grants VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Y. Xu was supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond” No. 101020331. article_processing_charge: No article_type: original author: - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 - first_name: Yuanyuan full_name: Xu, Yuanyuan id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3 last_name: Xu orcid: 0000-0003-1559-1205 citation: ama: Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 2023;33(1):677-725. doi:10.1214/22-aap1826 apa: Schnelli, K., & Xu, Y. (2023). Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1826 chicago: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1826. ieee: K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices,” The Annals of Applied Probability, vol. 33, no. 1. Institute of Mathematical Statistics, pp. 677–725, 2023. ista: Schnelli K, Xu Y. 2023. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 33(1), 677–725. mla: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied Probability, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp. 677–725, doi:10.1214/22-aap1826. short: K. Schnelli, Y. Xu, The Annals of Applied Probability 33 (2023) 677–725. date_created: 2024-01-10T09:23:31Z date_published: 2023-02-01T00:00:00Z date_updated: 2024-01-10T13:31:46Z day: '01' department: - _id: LaEr doi: 10.1214/22-aap1826 ec_funded: 1 external_id: arxiv: - '2108.02728' isi: - '000946432400021' intvolume: ' 33' isi: 1 issue: '1' keyword: - Statistics - Probability and Uncertainty - Statistics and Probability language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2108.02728 month: '02' oa: 1 oa_version: Preprint page: 677-725 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: The Annals of Applied Probability publication_identifier: issn: - 1050-5164 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' scopus_import: '1' status: public title: Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 33 year: '2023' ... --- _id: '14780' abstract: - lang: eng text: In this paper, we study the eigenvalues and eigenvectors of the spiked invariant multiplicative models when the randomness is from Haar matrices. We establish the limits of the outlier eigenvalues λˆi and the generalized components (⟨v,uˆi⟩ for any deterministic vector v) of the outlier eigenvectors uˆi with optimal convergence rates. Moreover, we prove that the non-outlier eigenvalues stick with those of the unspiked matrices and the non-outlier eigenvectors are delocalized. The results also hold near the so-called BBP transition and for degenerate spikes. On one hand, our results can be regarded as a refinement of the counterparts of [12] under additional regularity conditions. On the other hand, they can be viewed as an analog of [34] by replacing the random matrix with i.i.d. entries with Haar random matrix. acknowledgement: The authors would like to thank the editor, the associated editor and two anonymous referees for their many critical suggestions which have significantly improved the paper. The authors are also grateful to Zhigang Bao and Ji Oon Lee for many helpful discussions. The first author also wants to thank Hari Bercovici for many useful comments. The first author is partially supported by National Science Foundation DMS-2113489 and the second author is supported by ERC Advanced Grant “RMTBeyond” No. 101020331. article_processing_charge: Yes (in subscription journal) article_type: original author: - first_name: Xiucai full_name: Ding, Xiucai last_name: Ding - first_name: Hong Chang full_name: Ji, Hong Chang id: dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d last_name: Ji citation: ama: Ding X, Ji HC. Spiked multiplicative random matrices and principal components. Stochastic Processes and their Applications. 2023;163:25-60. doi:10.1016/j.spa.2023.05.009 apa: Ding, X., & Ji, H. C. (2023). Spiked multiplicative random matrices and principal components. Stochastic Processes and Their Applications. Elsevier. https://doi.org/10.1016/j.spa.2023.05.009 chicago: Ding, Xiucai, and Hong Chang Ji. “Spiked Multiplicative Random Matrices and Principal Components.” Stochastic Processes and Their Applications. Elsevier, 2023. https://doi.org/10.1016/j.spa.2023.05.009. ieee: X. Ding and H. C. Ji, “Spiked multiplicative random matrices and principal components,” Stochastic Processes and their Applications, vol. 163. Elsevier, pp. 25–60, 2023. ista: Ding X, Ji HC. 2023. Spiked multiplicative random matrices and principal components. Stochastic Processes and their Applications. 163, 25–60. mla: Ding, Xiucai, and Hong Chang Ji. “Spiked Multiplicative Random Matrices and Principal Components.” Stochastic Processes and Their Applications, vol. 163, Elsevier, 2023, pp. 25–60, doi:10.1016/j.spa.2023.05.009. short: X. Ding, H.C. Ji, Stochastic Processes and Their Applications 163 (2023) 25–60. date_created: 2024-01-10T09:29:25Z date_published: 2023-09-01T00:00:00Z date_updated: 2024-01-16T08:49:51Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1016/j.spa.2023.05.009 ec_funded: 1 external_id: arxiv: - '2302.13502' isi: - '001113615900001' file: - access_level: open_access checksum: 46a708b0cd5569a73d0f3d6c3e0a44dc content_type: application/pdf creator: dernst date_created: 2024-01-16T08:47:31Z date_updated: 2024-01-16T08:47:31Z file_id: '14806' file_name: 2023_StochasticProcAppl_Ding.pdf file_size: 1870349 relation: main_file success: 1 file_date_updated: 2024-01-16T08:47:31Z has_accepted_license: '1' intvolume: ' 163' isi: 1 keyword: - Applied Mathematics - Modeling and Simulation - Statistics and Probability language: - iso: eng month: '09' oa: 1 oa_version: Published Version page: 25-60 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Stochastic Processes and their Applications publication_identifier: eissn: - 1879-209X issn: - 0304-4149 publication_status: published publisher: Elsevier quality_controlled: '1' status: public title: Spiked multiplicative random matrices and principal components tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 163 year: '2023' ... --- _id: '14849' abstract: - lang: eng text: We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an n×n random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal. acknowledgement: "The second and the fourth author were supported by the ERC Advanced Grant\r\n“RMTBeyond” No. 101020331. The third author was supported by Dr. Max Rössler, the\r\nWalter Haefner Foundation and the ETH Zürich Foundation." article_processing_charge: No article_type: original author: - first_name: Giorgio full_name: Cipolloni, Giorgio id: 42198EFA-F248-11E8-B48F-1D18A9856A87 last_name: Cipolloni orcid: 0000-0002-4901-7992 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Dominik J full_name: Schröder, Dominik J id: 408ED176-F248-11E8-B48F-1D18A9856A87 last_name: Schröder orcid: 0000-0002-2904-1856 - first_name: Yuanyuan full_name: Xu, Yuanyuan last_name: Xu citation: ama: Cipolloni G, Erdös L, Schröder DJ, Xu Y. On the rightmost eigenvalue of non-Hermitian random matrices. The Annals of Probability. 2023;51(6):2192-2242. doi:10.1214/23-aop1643 apa: Cipolloni, G., Erdös, L., Schröder, D. J., & Xu, Y. (2023). On the rightmost eigenvalue of non-Hermitian random matrices. The Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/23-aop1643 chicago: Cipolloni, Giorgio, László Erdös, Dominik J Schröder, and Yuanyuan Xu. “On the Rightmost Eigenvalue of Non-Hermitian Random Matrices.” The Annals of Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/23-aop1643. ieee: G. Cipolloni, L. Erdös, D. J. Schröder, and Y. Xu, “On the rightmost eigenvalue of non-Hermitian random matrices,” The Annals of Probability, vol. 51, no. 6. Institute of Mathematical Statistics, pp. 2192–2242, 2023. ista: Cipolloni G, Erdös L, Schröder DJ, Xu Y. 2023. On the rightmost eigenvalue of non-Hermitian random matrices. The Annals of Probability. 51(6), 2192–2242. mla: Cipolloni, Giorgio, et al. “On the Rightmost Eigenvalue of Non-Hermitian Random Matrices.” The Annals of Probability, vol. 51, no. 6, Institute of Mathematical Statistics, 2023, pp. 2192–242, doi:10.1214/23-aop1643. short: G. Cipolloni, L. Erdös, D.J. Schröder, Y. Xu, The Annals of Probability 51 (2023) 2192–2242. date_created: 2024-01-22T08:08:41Z date_published: 2023-11-01T00:00:00Z date_updated: 2024-01-23T10:56:30Z day: '01' department: - _id: LaEr doi: 10.1214/23-aop1643 ec_funded: 1 external_id: arxiv: - '2206.04448' intvolume: ' 51' issue: '6' keyword: - Statistics - Probability and Uncertainty - Statistics and Probability language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2206.04448 month: '11' oa: 1 oa_version: Preprint page: 2192-2242 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: The Annals of Probability publication_identifier: issn: - 0091-1798 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' status: public title: On the rightmost eigenvalue of non-Hermitian random matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 51 year: '2023' ... --- _id: '15128' abstract: - lang: eng text: "We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions. The main novel ingredient is an optimal local law for the two-point function $T(z,\\zeta)$ and a general class of related quantities involving two resolvents\r\nat nearby spectral parameters. " acknowledgement: Supported by the ERC Advanced Grant ”RMTBeyond” No. 101020331 article_number: '2301.01712' article_processing_charge: No author: - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Riabov V. Mesoscopic eigenvalue statistics for Wigner-type matrices. arXiv. doi:10.48550/arXiv.2301.01712 apa: Riabov, V. (n.d.). Mesoscopic eigenvalue statistics for Wigner-type matrices. arXiv. https://doi.org/10.48550/arXiv.2301.01712 chicago: Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2301.01712. ieee: V. Riabov, “Mesoscopic eigenvalue statistics for Wigner-type matrices,” arXiv. . ista: Riabov V. Mesoscopic eigenvalue statistics for Wigner-type matrices. arXiv, 2301.01712. mla: Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.” ArXiv, 2301.01712, doi:10.48550/arXiv.2301.01712. short: V. Riabov, ArXiv (n.d.). date_created: 2024-03-20T09:41:04Z date_published: 2023-01-04T00:00:00Z date_updated: 2024-03-25T12:48:20Z day: '04' department: - _id: GradSch - _id: LaEr doi: 10.48550/arXiv.2301.01712 ec_funded: 1 external_id: arxiv: - '2301.01712' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2301.01712 month: '01' oa: 1 oa_version: Preprint project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: arXiv publication_status: submitted status: public title: Mesoscopic eigenvalue statistics for Wigner-type matrices type: preprint user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2023' ... --- _id: '12179' abstract: - lang: eng text: We derive an accurate lower tail estimate on the lowest singular value σ1(X−z) of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z. Such shift effectively changes the upper tail behavior of the condition number κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away from the real axis. This sharpens and resolves a recent conjecture in [J. Banks et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys., 1 (2020), pp. 101--146]. article_processing_charge: No article_type: original author: - first_name: Giorgio full_name: Cipolloni, Giorgio id: 42198EFA-F248-11E8-B48F-1D18A9856A87 last_name: Cipolloni orcid: 0000-0002-4901-7992 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Dominik J full_name: Schröder, Dominik J id: 408ED176-F248-11E8-B48F-1D18A9856A87 last_name: Schröder orcid: 0000-0002-2904-1856 citation: ama: Cipolloni G, Erdös L, Schröder DJ. On the condition number of the shifted real Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications. 2022;43(3):1469-1487. doi:10.1137/21m1424408 apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2022). On the condition number of the shifted real Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/21m1424408 chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “On the Condition Number of the Shifted Real Ginibre Ensemble.” SIAM Journal on Matrix Analysis and Applications. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21m1424408. ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “On the condition number of the shifted real Ginibre ensemble,” SIAM Journal on Matrix Analysis and Applications, vol. 43, no. 3. Society for Industrial and Applied Mathematics, pp. 1469–1487, 2022. ista: Cipolloni G, Erdös L, Schröder DJ. 2022. On the condition number of the shifted real Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications. 43(3), 1469–1487. mla: Cipolloni, Giorgio, et al. “On the Condition Number of the Shifted Real Ginibre Ensemble.” SIAM Journal on Matrix Analysis and Applications, vol. 43, no. 3, Society for Industrial and Applied Mathematics, 2022, pp. 1469–87, doi:10.1137/21m1424408. short: G. Cipolloni, L. Erdös, D.J. Schröder, SIAM Journal on Matrix Analysis and Applications 43 (2022) 1469–1487. date_created: 2023-01-12T12:12:38Z date_published: 2022-07-01T00:00:00Z date_updated: 2023-01-27T06:56:06Z day: '01' department: - _id: LaEr doi: 10.1137/21m1424408 external_id: arxiv: - '2105.13719' intvolume: ' 43' issue: '3' keyword: - Analysis language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2105.13719 month: '07' oa: 1 oa_version: Preprint page: 1469-1487 publication: SIAM Journal on Matrix Analysis and Applications publication_identifier: eissn: - 1095-7162 issn: - 0895-4798 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: On the condition number of the shifted real Ginibre ensemble type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 43 year: '2022' ...