--- _id: '429' abstract: - lang: eng text: We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent. acknowledgement: "Open access funding provided by Institute of Science and Technology (IST Austria).\r\n" article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Oskari H full_name: Ajanki, Oskari H id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87 last_name: Ajanki - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Torben H full_name: Krüger, Torben H id: 3020C786-F248-11E8-B48F-1D18A9856A87 last_name: Krüger orcid: 0000-0002-4821-3297 citation: ama: Ajanki OH, Erdös L, Krüger TH. Stability of the matrix Dyson equation and random matrices with correlations. Probability Theory and Related Fields. 2019;173(1-2):293–373. doi:10.1007/s00440-018-0835-z apa: Ajanki, O. H., Erdös, L., & Krüger, T. H. (2019). Stability of the matrix Dyson equation and random matrices with correlations. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-018-0835-z chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Stability of the Matrix Dyson Equation and Random Matrices with Correlations.” Probability Theory and Related Fields. Springer, 2019. https://doi.org/10.1007/s00440-018-0835-z. ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Stability of the matrix Dyson equation and random matrices with correlations,” Probability Theory and Related Fields, vol. 173, no. 1–2. Springer, pp. 293–373, 2019. ista: Ajanki OH, Erdös L, Krüger TH. 2019. Stability of the matrix Dyson equation and random matrices with correlations. Probability Theory and Related Fields. 173(1–2), 293–373. mla: Ajanki, Oskari H., et al. “Stability of the Matrix Dyson Equation and Random Matrices with Correlations.” Probability Theory and Related Fields, vol. 173, no. 1–2, Springer, 2019, pp. 293–373, doi:10.1007/s00440-018-0835-z. short: O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 173 (2019) 293–373. date_created: 2018-12-11T11:46:25Z date_published: 2019-02-01T00:00:00Z date_updated: 2023-08-24T14:39:00Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s00440-018-0835-z ec_funded: 1 external_id: isi: - '000459396500007' file: - access_level: open_access checksum: f9354fa5c71f9edd17132588f0dc7d01 content_type: application/pdf creator: dernst date_created: 2018-12-17T16:12:08Z date_updated: 2020-07-14T12:46:26Z file_id: '5720' file_name: 2018_ProbTheory_Ajanki.pdf file_size: 1201840 relation: main_file file_date_updated: 2020-07-14T12:46:26Z has_accepted_license: '1' intvolume: ' 173' isi: 1 issue: 1-2 language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '02' oa: 1 oa_version: Published Version page: 293–373 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Probability Theory and Related Fields publication_identifier: eissn: - '14322064' issn: - '01788051' publication_status: published publisher: Springer publist_id: '7394' quality_controlled: '1' scopus_import: '1' status: public title: Stability of the matrix Dyson equation and random matrices with correlations 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: 173 year: '2019' ... --- _id: '721' abstract: - lang: eng text: 'Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.' author: - first_name: Oskari H full_name: Ajanki, Oskari H id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87 last_name: Ajanki - first_name: Torben H full_name: Krüger, Torben H id: 3020C786-F248-11E8-B48F-1D18A9856A87 last_name: Krüger orcid: 0000-0002-4821-3297 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 citation: ama: Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 2017;70(9):1672-1705. doi:10.1002/cpa.21639 apa: Ajanki, O. H., Krüger, T. H., & Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. Wiley-Blackwell. https://doi.org/10.1002/cpa.21639 chicago: Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” Communications on Pure and Applied Mathematics. Wiley-Blackwell, 2017. https://doi.org/10.1002/cpa.21639. ieee: O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” Communications on Pure and Applied Mathematics, vol. 70, no. 9. Wiley-Blackwell, pp. 1672–1705, 2017. ista: Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 70(9), 1672–1705. mla: Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” Communications on Pure and Applied Mathematics, vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi:10.1002/cpa.21639. short: O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705. date_created: 2018-12-11T11:48:08Z date_published: 2017-09-01T00:00:00Z date_updated: 2021-01-12T08:12:24Z day: '01' department: - _id: LaEr doi: 10.1002/cpa.21639 ec_funded: 1 intvolume: ' 70' issue: '9' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1512.03703 month: '09' oa: 1 oa_version: Submitted Version page: 1672 - 1705 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Communications on Pure and Applied Mathematics publication_identifier: issn: - '00103640' publication_status: published publisher: Wiley-Blackwell publist_id: '6959' quality_controlled: '1' scopus_import: 1 status: public title: Singularities of solutions to quadratic vector equations on the complex upper half plane type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 70 year: '2017' ... --- _id: '1337' abstract: - lang: eng text: We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes. acknowledgement: 'Open access funding provided by Institute of Science and Technology (IST Austria). ' article_processing_charge: Yes (via OA deal) author: - first_name: Oskari H full_name: Ajanki, Oskari H id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87 last_name: Ajanki - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Torben H full_name: Krüger, Torben H id: 3020C786-F248-11E8-B48F-1D18A9856A87 last_name: Krüger orcid: 0000-0002-4821-3297 citation: ama: Ajanki OH, Erdös L, Krüger TH. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 2017;169(3-4):667-727. doi:10.1007/s00440-016-0740-2 apa: Ajanki, O. H., Erdös, L., & Krüger, T. H. (2017). Universality for general Wigner-type matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-016-0740-2 chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Universality for General Wigner-Type Matrices.” Probability Theory and Related Fields. Springer, 2017. https://doi.org/10.1007/s00440-016-0740-2. ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Universality for general Wigner-type matrices,” Probability Theory and Related Fields, vol. 169, no. 3–4. Springer, pp. 667–727, 2017. ista: Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 169(3–4), 667–727. mla: Ajanki, Oskari H., et al. “Universality for General Wigner-Type Matrices.” Probability Theory and Related Fields, vol. 169, no. 3–4, Springer, 2017, pp. 667–727, doi:10.1007/s00440-016-0740-2. short: O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 169 (2017) 667–727. date_created: 2018-12-11T11:51:27Z date_published: 2017-12-01T00:00:00Z date_updated: 2023-09-20T11:14:17Z day: '01' ddc: - '510' - '530' department: - _id: LaEr doi: 10.1007/s00440-016-0740-2 ec_funded: 1 external_id: isi: - '000414358400002' file: - access_level: open_access checksum: 29f5a72c3f91e408aeb9e78344973803 content_type: application/pdf creator: system date_created: 2018-12-12T10:08:25Z date_updated: 2020-07-14T12:44:44Z file_id: '4686' file_name: IST-2017-657-v1+2_s00440-016-0740-2.pdf file_size: 988843 relation: main_file file_date_updated: 2020-07-14T12:44:44Z has_accepted_license: '1' intvolume: ' 169' isi: 1 issue: 3-4 language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 667 - 727 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Probability Theory and Related Fields publication_identifier: issn: - '01788051' publication_status: published publisher: Springer publist_id: '5930' pubrep_id: '657' quality_controlled: '1' scopus_import: '1' status: public title: Universality for general Wigner-type 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: 169 year: '2017' ... --- _id: '1489' abstract: - lang: eng text: 'We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries. ' acknowledgement: Open access funding provided by Institute of Science and Technology (IST Austria). Oskari H. Ajanki was Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12 Grant of the German Research Council. László Erdős was Partially supported by ERC Advanced Grant RANMAT No. 338804. Torben Krüger was Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12 Grant of the German Research Council. article_processing_charge: Yes (via OA deal) author: - first_name: Oskari H full_name: Ajanki, Oskari H id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87 last_name: Ajanki - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Torben H full_name: Krüger, Torben H id: 3020C786-F248-11E8-B48F-1D18A9856A87 last_name: Krüger orcid: 0000-0002-4821-3297 citation: ama: Ajanki OH, Erdös L, Krüger TH. Local spectral statistics of Gaussian matrices with correlated entries. Journal of Statistical Physics. 2016;163(2):280-302. doi:10.1007/s10955-016-1479-y apa: Ajanki, O. H., Erdös, L., & Krüger, T. H. (2016). Local spectral statistics of Gaussian matrices with correlated entries. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-016-1479-y chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Spectral Statistics of Gaussian Matrices with Correlated Entries.” Journal of Statistical Physics. Springer, 2016. https://doi.org/10.1007/s10955-016-1479-y. ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local spectral statistics of Gaussian matrices with correlated entries,” Journal of Statistical Physics, vol. 163, no. 2. Springer, pp. 280–302, 2016. ista: Ajanki OH, Erdös L, Krüger TH. 2016. Local spectral statistics of Gaussian matrices with correlated entries. Journal of Statistical Physics. 163(2), 280–302. mla: Ajanki, Oskari H., et al. “Local Spectral Statistics of Gaussian Matrices with Correlated Entries.” Journal of Statistical Physics, vol. 163, no. 2, Springer, 2016, pp. 280–302, doi:10.1007/s10955-016-1479-y. short: O.H. Ajanki, L. Erdös, T.H. Krüger, Journal of Statistical Physics 163 (2016) 280–302. date_created: 2018-12-11T11:52:19Z date_published: 2016-04-01T00:00:00Z date_updated: 2021-01-12T06:51:05Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s10955-016-1479-y ec_funded: 1 file: - access_level: open_access checksum: 7139598dcb1cafbe6866bd2bfd732b33 content_type: application/pdf creator: system date_created: 2018-12-12T10:11:16Z date_updated: 2020-07-14T12:44:57Z file_id: '4869' file_name: IST-2016-516-v1+1_s10955-016-1479-y.pdf file_size: 660602 relation: main_file file_date_updated: 2020-07-14T12:44:57Z has_accepted_license: '1' intvolume: ' 163' issue: '2' language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: 280 - 302 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Journal of Statistical Physics publication_status: published publisher: Springer publist_id: '5698' pubrep_id: '516' quality_controlled: '1' scopus_import: 1 status: public title: Local spectral statistics of Gaussian matrices with correlated entries 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: '2016' ... --- _id: '2179' abstract: - lang: eng text: We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 to the case when the matrix of variances has an eigenvalue -1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where the variances of the entries of X may vary. author: - first_name: Oskari H full_name: Ajanki, Oskari H id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87 last_name: Ajanki - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Torben H full_name: Krüger, Torben H id: 3020C786-F248-11E8-B48F-1D18A9856A87 last_name: Krüger orcid: 0000-0002-4821-3297 citation: ama: Ajanki OH, Erdös L, Krüger TH. Local semicircle law with imprimitive variance matrix. Electronic Communications in Probability. 2014;19. doi:10.1214/ECP.v19-3121 apa: Ajanki, O. H., Erdös, L., & Krüger, T. H. (2014). Local semicircle law with imprimitive variance matrix. Electronic Communications in Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/ECP.v19-3121 chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Semicircle Law with Imprimitive Variance Matrix.” Electronic Communications in Probability. Institute of Mathematical Statistics, 2014. https://doi.org/10.1214/ECP.v19-3121. ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local semicircle law with imprimitive variance matrix,” Electronic Communications in Probability, vol. 19. Institute of Mathematical Statistics, 2014. ista: Ajanki OH, Erdös L, Krüger TH. 2014. Local semicircle law with imprimitive variance matrix. Electronic Communications in Probability. 19. mla: Ajanki, Oskari H., et al. “Local Semicircle Law with Imprimitive Variance Matrix.” Electronic Communications in Probability, vol. 19, Institute of Mathematical Statistics, 2014, doi:10.1214/ECP.v19-3121. short: O.H. Ajanki, L. Erdös, T.H. Krüger, Electronic Communications in Probability 19 (2014). date_created: 2018-12-11T11:56:10Z date_published: 2014-06-09T00:00:00Z date_updated: 2021-01-12T06:55:48Z day: '09' ddc: - '570' department: - _id: LaEr doi: 10.1214/ECP.v19-3121 file: - access_level: open_access checksum: bd8a041c76d62fe820bf73ff13ce7d1b content_type: application/pdf creator: system date_created: 2018-12-12T10:09:06Z date_updated: 2020-07-14T12:45:31Z file_id: '4729' file_name: IST-2016-426-v1+1_3121-17518-1-PB.pdf file_size: 327322 relation: main_file file_date_updated: 2020-07-14T12:45:31Z has_accepted_license: '1' intvolume: ' 19' language: - iso: eng month: '06' oa: 1 oa_version: Published Version publication: Electronic Communications in Probability publication_status: published publisher: Institute of Mathematical Statistics publist_id: '4803' pubrep_id: '426' quality_controlled: '1' scopus_import: 1 status: public title: Local semicircle law with imprimitive variance matrix 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: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 19 year: '2014' ...