--- _id: '9550' abstract: - lang: eng text: 'We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. ' acknowledgement: The first author is supported in part by Hong Kong RGC Grant GRF 16301519 and NSFC 11871425. The second author is supported in part by ERC Advanced Grant RANMAT 338804. The third author is supported in part by Swedish Research Council Grant VR-2017-05195 and the Knut and Alice Wallenberg Foundation article_number: e44 article_processing_charge: No article_type: original author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Bao Z, Erdös L, Schnelli K. Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. 2021;9. doi:10.1017/fms.2021.38 apa: Bao, Z., Erdös, L., & Schnelli, K. (2021). Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. Cambridge University Press. https://doi.org/10.1017/fms.2021.38 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “Equipartition Principle for Wigner Matrices.” Forum of Mathematics, Sigma. Cambridge University Press, 2021. https://doi.org/10.1017/fms.2021.38. ieee: Z. Bao, L. Erdös, and K. Schnelli, “Equipartition principle for Wigner matrices,” Forum of Mathematics, Sigma, vol. 9. Cambridge University Press, 2021. ista: Bao Z, Erdös L, Schnelli K. 2021. Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. 9, e44. mla: Bao, Zhigang, et al. “Equipartition Principle for Wigner Matrices.” Forum of Mathematics, Sigma, vol. 9, e44, Cambridge University Press, 2021, doi:10.1017/fms.2021.38. short: Z. Bao, L. Erdös, K. Schnelli, Forum of Mathematics, Sigma 9 (2021). date_created: 2021-06-13T22:01:33Z date_published: 2021-05-27T00:00:00Z date_updated: 2023-08-08T14:03:40Z day: '27' ddc: - '510' department: - _id: LaEr doi: 10.1017/fms.2021.38 ec_funded: 1 external_id: arxiv: - '2008.07061' isi: - '000654960800001' file: - access_level: open_access checksum: 47c986578de132200d41e6d391905519 content_type: application/pdf creator: cziletti date_created: 2021-06-15T14:40:45Z date_updated: 2021-06-15T14:40:45Z file_id: '9555' file_name: 2021_ForumMath_Bao.pdf file_size: 483458 relation: main_file success: 1 file_date_updated: 2021-06-15T14:40:45Z has_accepted_license: '1' intvolume: ' 9' isi: 1 language: - iso: eng month: '05' oa: 1 oa_version: Published Version project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Forum of Mathematics, Sigma publication_identifier: eissn: - '20505094' publication_status: published publisher: Cambridge University Press quality_controlled: '1' scopus_import: '1' status: public title: Equipartition principle for Wigner matrices 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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 9 year: '2021' ... --- _id: '9104' abstract: - lang: eng text: We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5]. acknowledgement: "Supported in part by Hong Kong RGC Grant ECS 26301517.\r\nSupported in part by ERC Advanced Grant RANMAT No. 338804.\r\nSupported in part by the Knut and Alice Wallenberg Foundation and the Swedish Research Council Grant VR-2017-05195." article_processing_charge: No article_type: original author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Bao Z, Erdös L, Schnelli K. On the support of the free additive convolution. Journal d’Analyse Mathematique. 2020;142:323-348. doi:10.1007/s11854-020-0135-2 apa: Bao, Z., Erdös, L., & Schnelli, K. (2020). On the support of the free additive convolution. Journal d’Analyse Mathematique. Springer Nature. https://doi.org/10.1007/s11854-020-0135-2 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “On the Support of the Free Additive Convolution.” Journal d’Analyse Mathematique. Springer Nature, 2020. https://doi.org/10.1007/s11854-020-0135-2. ieee: Z. Bao, L. Erdös, and K. Schnelli, “On the support of the free additive convolution,” Journal d’Analyse Mathematique, vol. 142. Springer Nature, pp. 323–348, 2020. ista: Bao Z, Erdös L, Schnelli K. 2020. On the support of the free additive convolution. Journal d’Analyse Mathematique. 142, 323–348. mla: Bao, Zhigang, et al. “On the Support of the Free Additive Convolution.” Journal d’Analyse Mathematique, vol. 142, Springer Nature, 2020, pp. 323–48, doi:10.1007/s11854-020-0135-2. short: Z. Bao, L. Erdös, K. Schnelli, Journal d’Analyse Mathematique 142 (2020) 323–348. date_created: 2021-02-07T23:01:15Z date_published: 2020-11-01T00:00:00Z date_updated: 2023-08-24T11:16:03Z day: '01' department: - _id: LaEr doi: 10.1007/s11854-020-0135-2 ec_funded: 1 external_id: arxiv: - '1804.11199' isi: - '000611879400008' intvolume: ' 142' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1804.11199 month: '11' oa: 1 oa_version: Preprint page: 323-348 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Journal d'Analyse Mathematique publication_identifier: eissn: - '15658538' issn: - '00217670' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: On the support of the free additive convolution type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 142 year: '2020' ... --- _id: '10862' abstract: - lang: eng text: We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4], [5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix. acknowledgement: Partially supported by ERC Advanced Grant RANMAT No. 338804. article_number: '108639' article_processing_charge: No article_type: original author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin last_name: Schnelli citation: ama: Bao Z, Erdös L, Schnelli K. Spectral rigidity for addition of random matrices at the regular edge. Journal of Functional Analysis. 2020;279(7). doi:10.1016/j.jfa.2020.108639 apa: Bao, Z., Erdös, L., & Schnelli, K. (2020). Spectral rigidity for addition of random matrices at the regular edge. Journal of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2020.108639 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “Spectral Rigidity for Addition of Random Matrices at the Regular Edge.” Journal of Functional Analysis. Elsevier, 2020. https://doi.org/10.1016/j.jfa.2020.108639. ieee: Z. Bao, L. Erdös, and K. Schnelli, “Spectral rigidity for addition of random matrices at the regular edge,” Journal of Functional Analysis, vol. 279, no. 7. Elsevier, 2020. ista: Bao Z, Erdös L, Schnelli K. 2020. Spectral rigidity for addition of random matrices at the regular edge. Journal of Functional Analysis. 279(7), 108639. mla: Bao, Zhigang, et al. “Spectral Rigidity for Addition of Random Matrices at the Regular Edge.” Journal of Functional Analysis, vol. 279, no. 7, 108639, Elsevier, 2020, doi:10.1016/j.jfa.2020.108639. short: Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 279 (2020). date_created: 2022-03-18T10:18:59Z date_published: 2020-10-15T00:00:00Z date_updated: 2023-08-24T14:08:42Z day: '15' department: - _id: LaEr doi: 10.1016/j.jfa.2020.108639 ec_funded: 1 external_id: arxiv: - '1708.01597' isi: - '000559623200009' intvolume: ' 279' isi: 1 issue: '7' keyword: - Analysis language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1708.01597 month: '10' oa: 1 oa_version: Preprint project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Journal of Functional Analysis publication_identifier: issn: - 0022-1236 publication_status: published publisher: Elsevier quality_controlled: '1' scopus_import: '1' status: public title: Spectral rigidity for addition of random matrices at the regular edge type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 279 year: '2020' ... --- _id: '6511' abstract: - lang: eng text: Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N). article_processing_charge: No author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Bao Z, Erdös L, Schnelli K. Local single ring theorem on optimal scale. Annals of Probability. 2019;47(3):1270-1334. doi:10.1214/18-AOP1284 apa: Bao, Z., Erdös, L., & Schnelli, K. (2019). Local single ring theorem on optimal scale. Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/18-AOP1284 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Single Ring Theorem on Optimal Scale.” Annals of Probability. Institute of Mathematical Statistics, 2019. https://doi.org/10.1214/18-AOP1284. ieee: Z. Bao, L. Erdös, and K. Schnelli, “Local single ring theorem on optimal scale,” Annals of Probability, vol. 47, no. 3. Institute of Mathematical Statistics, pp. 1270–1334, 2019. ista: Bao Z, Erdös L, Schnelli K. 2019. Local single ring theorem on optimal scale. Annals of Probability. 47(3), 1270–1334. mla: Bao, Zhigang, et al. “Local Single Ring Theorem on Optimal Scale.” Annals of Probability, vol. 47, no. 3, Institute of Mathematical Statistics, 2019, pp. 1270–334, doi:10.1214/18-AOP1284. short: Z. Bao, L. Erdös, K. Schnelli, Annals of Probability 47 (2019) 1270–1334. date_created: 2019-06-02T21:59:13Z date_published: 2019-05-01T00:00:00Z date_updated: 2023-08-28T09:32:29Z day: '01' department: - _id: LaEr doi: 10.1214/18-AOP1284 ec_funded: 1 external_id: arxiv: - '1612.05920' isi: - '000466616100003' intvolume: ' 47' isi: 1 issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1612.05920 month: '05' oa: 1 oa_version: Preprint page: 1270-1334 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Annals of Probability publication_identifier: issn: - '00911798' publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' scopus_import: '1' status: public title: Local single ring theorem on optimal scale type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 47 year: '2019' ... --- _id: '1528' abstract: - lang: eng text: 'We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.' acknowledgement: "Z. Bao was supported by ERC Advanced Grant RANMAT No. 338804; L. Erdős was partially supported by ERC Advanced Grant RANMAT No. 338804.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). The authors are very grateful to the anonymous referees for careful reading and valuable comments, which helped to improve the organization." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 citation: ama: Bao Z, Erdös L. Delocalization for a class of random block band matrices. Probability Theory and Related Fields. 2017;167(3-4):673-776. doi:10.1007/s00440-015-0692-y apa: Bao, Z., & Erdös, L. (2017). Delocalization for a class of random block band matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-015-0692-y chicago: Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” Probability Theory and Related Fields. Springer, 2017. https://doi.org/10.1007/s00440-015-0692-y. ieee: Z. Bao and L. Erdös, “Delocalization for a class of random block band matrices,” Probability Theory and Related Fields, vol. 167, no. 3–4. Springer, pp. 673–776, 2017. ista: Bao Z, Erdös L. 2017. Delocalization for a class of random block band matrices. Probability Theory and Related Fields. 167(3–4), 673–776. mla: Bao, Zhigang, and László Erdös. “Delocalization for a Class of Random Block Band Matrices.” Probability Theory and Related Fields, vol. 167, no. 3–4, Springer, 2017, pp. 673–776, doi:10.1007/s00440-015-0692-y. short: Z. Bao, L. Erdös, Probability Theory and Related Fields 167 (2017) 673–776. date_created: 2018-12-11T11:52:32Z date_published: 2017-04-01T00:00:00Z date_updated: 2023-09-20T09:42:12Z day: '01' ddc: - '530' department: - _id: LaEr doi: 10.1007/s00440-015-0692-y ec_funded: 1 external_id: isi: - '000398842700004' file: - access_level: open_access checksum: 67afa85ff1e220cbc1f9f477a828513c content_type: application/pdf creator: system date_created: 2018-12-12T10:08:05Z date_updated: 2020-07-14T12:45:00Z file_id: '4665' file_name: IST-2016-489-v1+1_s00440-015-0692-y.pdf file_size: 1615755 relation: main_file file_date_updated: 2020-07-14T12:45:00Z has_accepted_license: '1' intvolume: ' 167' isi: 1 issue: 3-4 language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: 673 - 776 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Probability Theory and Related Fields publication_identifier: issn: - '01788051' publication_status: published publisher: Springer publist_id: '5644' pubrep_id: '489' quality_controlled: '1' scopus_import: '1' status: public title: Delocalization for a class of random block band matrices 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: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 167 year: '2017' ... --- _id: '1207' abstract: - lang: eng text: The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix. article_processing_charge: Yes (via OA deal) author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Bao Z, Erdös L, Schnelli K. Local law of addition of random matrices on optimal scale. Communications in Mathematical Physics. 2017;349(3):947-990. doi:10.1007/s00220-016-2805-6 apa: Bao, Z., Erdös, L., & Schnelli, K. (2017). Local law of addition of random matrices on optimal scale. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-016-2805-6 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Law of Addition of Random Matrices on Optimal Scale.” Communications in Mathematical Physics. Springer, 2017. https://doi.org/10.1007/s00220-016-2805-6. ieee: Z. Bao, L. Erdös, and K. Schnelli, “Local law of addition of random matrices on optimal scale,” Communications in Mathematical Physics, vol. 349, no. 3. Springer, pp. 947–990, 2017. ista: Bao Z, Erdös L, Schnelli K. 2017. Local law of addition of random matrices on optimal scale. Communications in Mathematical Physics. 349(3), 947–990. mla: Bao, Zhigang, et al. “Local Law of Addition of Random Matrices on Optimal Scale.” Communications in Mathematical Physics, vol. 349, no. 3, Springer, 2017, pp. 947–90, doi:10.1007/s00220-016-2805-6. short: Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349 (2017) 947–990. date_created: 2018-12-11T11:50:43Z date_published: 2017-02-01T00:00:00Z date_updated: 2023-09-20T11:16:57Z day: '01' ddc: - '530' department: - _id: LaEr doi: 10.1007/s00220-016-2805-6 ec_funded: 1 external_id: isi: - '000393696700005' file: - access_level: open_access checksum: ddff79154c3daf27237de5383b1264a9 content_type: application/pdf creator: system date_created: 2018-12-12T10:14:47Z date_updated: 2020-07-14T12:44:39Z file_id: '5102' file_name: IST-2016-722-v1+1_s00220-016-2805-6.pdf file_size: 1033743 relation: main_file file_date_updated: 2020-07-14T12:44:39Z has_accepted_license: '1' intvolume: ' 349' isi: 1 issue: '3' language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: 947 - 990 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Communications in Mathematical Physics publication_identifier: issn: - '00103616' publication_status: published publisher: Springer publist_id: '6141' pubrep_id: '722' quality_controlled: '1' scopus_import: '1' status: public title: Local law of addition of random matrices on optimal scale 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: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 349 year: '2017' ... --- _id: '733' abstract: - lang: eng text: Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum. acknowledgement: Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation article_processing_charge: No author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 2017;319:251-291. doi:10.1016/j.aim.2017.08.028 apa: Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2017.08.028 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” Advances in Mathematics. Academic Press, 2017. https://doi.org/10.1016/j.aim.2017.08.028. ieee: Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” Advances in Mathematics, vol. 319. Academic Press, pp. 251–291, 2017. ista: Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291. mla: Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” Advances in Mathematics, vol. 319, Academic Press, 2017, pp. 251–91, doi:10.1016/j.aim.2017.08.028. short: Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291. date_created: 2018-12-11T11:48:13Z date_published: 2017-10-15T00:00:00Z date_updated: 2023-09-28T11:30:42Z day: '15' department: - _id: LaEr doi: 10.1016/j.aim.2017.08.028 ec_funded: 1 external_id: isi: - '000412150400010' intvolume: ' 319' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1606.03076 month: '10' oa: 1 oa_version: Submitted Version page: 251 - 291 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Advances in Mathematics publication_status: published publisher: Academic Press publist_id: '6935' quality_controlled: '1' scopus_import: '1' status: public title: Convergence rate for spectral distribution of addition of random matrices type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 319 year: '2017' ... --- _id: '1434' abstract: - lang: eng text: We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3. author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Bao Z, Erdös L, Schnelli K. Local stability of the free additive convolution. Journal of Functional Analysis. 2016;271(3):672-719. doi:10.1016/j.jfa.2016.04.006 apa: Bao, Z., Erdös, L., & Schnelli, K. (2016). Local stability of the free additive convolution. Journal of Functional Analysis. Academic Press. https://doi.org/10.1016/j.jfa.2016.04.006 chicago: Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Stability of the Free Additive Convolution.” Journal of Functional Analysis. Academic Press, 2016. https://doi.org/10.1016/j.jfa.2016.04.006. ieee: Z. Bao, L. Erdös, and K. Schnelli, “Local stability of the free additive convolution,” Journal of Functional Analysis, vol. 271, no. 3. Academic Press, pp. 672–719, 2016. ista: Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution. Journal of Functional Analysis. 271(3), 672–719. mla: Bao, Zhigang, et al. “Local Stability of the Free Additive Convolution.” Journal of Functional Analysis, vol. 271, no. 3, Academic Press, 2016, pp. 672–719, doi:10.1016/j.jfa.2016.04.006. short: Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016) 672–719. date_created: 2018-12-11T11:52:00Z date_published: 2016-08-01T00:00:00Z date_updated: 2021-01-12T06:50:42Z day: '01' department: - _id: LaEr doi: 10.1016/j.jfa.2016.04.006 ec_funded: 1 intvolume: ' 271' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1508.05905 month: '08' oa: 1 oa_version: Preprint page: 672 - 719 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Journal of Functional Analysis publication_status: published publisher: Academic Press publist_id: '5764' quality_controlled: '1' scopus_import: 1 status: public title: Local stability of the free additive convolution type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 271 year: '2016' ... --- _id: '1504' abstract: - lang: eng text: Let Q = (Q1, . . . , Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2, . . . , n}. Let Z = (Z1, . . . , Zn), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj , j = 1, . . . , n. Assume that Xi , i = 1, . . . ,p are i.i.d. copies of 1/√ p Z and X = Xp,n is the p × n random matrix with Xi as its ith row. Then Sn = XX is called the p × n Spearman's rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman's rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, p = p(n) and p/n→c ∈ (0,∞) as n→∞.We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann. Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavy-tailed ones. author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: Liang full_name: Lin, Liang last_name: Lin - first_name: Guangming full_name: Pan, Guangming last_name: Pan - first_name: Wang full_name: Zhou, Wang last_name: Zhou citation: ama: Bao Z, Lin L, Pan G, Zhou W. Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics. 2015;43(6):2588-2623. doi:10.1214/15-AOS1353 apa: Bao, Z., Lin, L., Pan, G., & Zhou, W. (2015). Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/15-AOS1353 chicago: Bao, Zhigang, Liang Lin, Guangming Pan, and Wang Zhou. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.” Annals of Statistics. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/15-AOS1353. ieee: Z. Bao, L. Lin, G. Pan, and W. Zhou, “Spectral statistics of large dimensional spearman s rank correlation matrix and its application,” Annals of Statistics, vol. 43, no. 6. Institute of Mathematical Statistics, pp. 2588–2623, 2015. ista: Bao Z, Lin L, Pan G, Zhou W. 2015. Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics. 43(6), 2588–2623. mla: Bao, Zhigang, et al. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.” Annals of Statistics, vol. 43, no. 6, Institute of Mathematical Statistics, 2015, pp. 2588–623, doi:10.1214/15-AOS1353. short: Z. Bao, L. Lin, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 2588–2623. date_created: 2018-12-11T11:52:24Z date_published: 2015-12-01T00:00:00Z date_updated: 2021-01-12T06:51:14Z day: '01' doi: 10.1214/15-AOS1353 extern: '1' intvolume: ' 43' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1312.5119 month: '12' oa: 1 oa_version: Published Version page: 2588 - 2623 publication: Annals of Statistics publication_status: published publisher: Institute of Mathematical Statistics publist_id: '5674' quality_controlled: '1' status: public title: Spectral statistics of large dimensional spearman s rank correlation matrix and its application type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 43 year: '2015' ... --- _id: '1505' abstract: - lang: eng text: This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij 's, we show that the limiting behavior of the largest eigenvalue of W N is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of W N converges weakly to the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of W N , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X . In summary, we establish the Tracy-Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on &Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed. acknowledgement: "B.Z. was supported in part by NSFC Grant 11071213, ZJNSF \ Grant R6090034 and SRFDP Grant 20100101110001. P.G. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11. Z.W. was supported \ in part by the Ministry of Education, Singapore, under Grant ARC 14/11, \ and by a Grant R-155-000-131-112 at the National University of Singapore\r\n" author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: Guangming full_name: Pan, Guangming last_name: Pan - first_name: Wang full_name: Zhou, Wang last_name: Zhou citation: ama: Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 2015;43(1):382-421. doi:10.1214/14-AOS1281 apa: Bao, Z., Pan, G., & Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/14-AOS1281 chicago: Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” Annals of Statistics. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/14-AOS1281. ieee: Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample covariance matrices with general population,” Annals of Statistics, vol. 43, no. 1. Institute of Mathematical Statistics, pp. 382–421, 2015. ista: Bao Z, Pan G, Zhou W. 2015. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 43(1), 382–421. mla: Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” Annals of Statistics, vol. 43, no. 1, Institute of Mathematical Statistics, 2015, pp. 382–421, doi:10.1214/14-AOS1281. short: Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421. date_created: 2018-12-11T11:52:25Z date_published: 2015-02-01T00:00:00Z date_updated: 2021-01-12T06:51:14Z day: '01' department: - _id: LaEr doi: 10.1214/14-AOS1281 intvolume: ' 43' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1304.5690 month: '02' oa: 1 oa_version: Preprint page: 382 - 421 publication: Annals of Statistics publication_status: published publisher: Institute of Mathematical Statistics publist_id: '5672' quality_controlled: '1' status: public title: Universality for the largest eigenvalue of sample covariance matrices with general population type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 43 year: '2015' ... --- _id: '1506' abstract: - lang: eng text: Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1). author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: Guangming full_name: Pan, Guangming last_name: Pan - first_name: Wang full_name: Zhou, Wang last_name: Zhou citation: ama: Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. Bernoulli. 2015;21(3):1600-1628. doi:10.3150/14-BEJ615 apa: Bao, Z., Pan, G., & Zhou, W. (2015). The logarithmic law of random determinant. Bernoulli. Bernoulli Society for Mathematical Statistics and Probability. https://doi.org/10.3150/14-BEJ615 chicago: Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random Determinant.” Bernoulli. Bernoulli Society for Mathematical Statistics and Probability, 2015. https://doi.org/10.3150/14-BEJ615. ieee: Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,” Bernoulli, vol. 21, no. 3. Bernoulli Society for Mathematical Statistics and Probability, pp. 1600–1628, 2015. ista: Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628. mla: Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” Bernoulli, vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability, 2015, pp. 1600–28, doi:10.3150/14-BEJ615. short: Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628. date_created: 2018-12-11T11:52:25Z date_published: 2015-08-01T00:00:00Z date_updated: 2021-01-12T06:51:14Z day: '01' department: - _id: LaEr doi: 10.3150/14-BEJ615 intvolume: ' 21' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1208.5823 month: '08' oa: 1 oa_version: Preprint page: 1600 - 1628 publication: Bernoulli publication_status: published publisher: Bernoulli Society for Mathematical Statistics and Probability publist_id: '5671' quality_controlled: '1' status: public title: The logarithmic law of random determinant type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 21 year: '2015' ... --- _id: '1585' abstract: - lang: eng text: In this paper, we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance. To this end, we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in Bai and Silverstein (2004). acknowledgement: "G. Pan was supported by MOE Tier 2 under Grant 2014-T2-2-060 and in part by Tier 1 under Grant RG25/14 through the Nanyang Technological University, Singapore. W. Zhou was supported by the National University of Singapore, Singapore, under Grant R-155-000-131-112.\r\n" author: - first_name: Zhigang full_name: Bao, Zhigang id: 442E6A6C-F248-11E8-B48F-1D18A9856A87 last_name: Bao orcid: 0000-0003-3036-1475 - first_name: Guangming full_name: Pan, Guangming last_name: Pan - first_name: Wang full_name: Zhou, Wang last_name: Zhou citation: ama: Bao Z, Pan G, Zhou W. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. 2015;61(6):3413-3426. doi:10.1109/TIT.2015.2421894 apa: Bao, Z., Pan, G., & Zhou, W. (2015). Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. IEEE. https://doi.org/10.1109/TIT.2015.2421894 chicago: Bao, Zhigang, Guangming Pan, and Wang Zhou. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” IEEE Transactions on Information Theory. IEEE, 2015. https://doi.org/10.1109/TIT.2015.2421894. ieee: Z. Bao, G. Pan, and W. Zhou, “Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices,” IEEE Transactions on Information Theory, vol. 61, no. 6. IEEE, pp. 3413–3426, 2015. ista: Bao Z, Pan G, Zhou W. 2015. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. 61(6), 3413–3426. mla: Bao, Zhigang, et al. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” IEEE Transactions on Information Theory, vol. 61, no. 6, IEEE, 2015, pp. 3413–26, doi:10.1109/TIT.2015.2421894. short: Z. Bao, G. Pan, W. Zhou, IEEE Transactions on Information Theory 61 (2015) 3413–3426. date_created: 2018-12-11T11:52:52Z date_published: 2015-06-01T00:00:00Z date_updated: 2021-01-12T06:51:46Z day: '01' department: - _id: LaEr doi: 10.1109/TIT.2015.2421894 intvolume: ' 61' issue: '6' language: - iso: eng month: '06' oa_version: None page: 3413 - 3426 publication: IEEE Transactions on Information Theory publication_status: published publisher: IEEE publist_id: '5586' quality_controlled: '1' scopus_import: 1 status: public title: Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 61 year: '2015' ...