--- _id: '615' abstract: - lang: eng text: We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion of the eigenvalues hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from Wigner's semicircle law. author: - 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: Erdös L, Schnelli K. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 2017;53(4):1606-1656. doi:10.1214/16-AIHP765 apa: Erdös, L., & Schnelli, K. (2017). Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AIHP765 chicago: Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” Annales de l’institut Henri Poincare (B) Probability and Statistics. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/16-AIHP765. ieee: L. Erdös and K. Schnelli, “Universality for random matrix flows with time dependent density,” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 53, no. 4. Institute of Mathematical Statistics, pp. 1606–1656, 2017. ista: Erdös L, Schnelli K. 2017. Universality for random matrix flows with time dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics. 53(4), 1606–1656. mla: Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with Time Dependent Density.” Annales de l’institut Henri Poincare (B) Probability and Statistics, vol. 53, no. 4, Institute of Mathematical Statistics, 2017, pp. 1606–56, doi:10.1214/16-AIHP765. short: L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656. date_created: 2018-12-11T11:47:30Z date_published: 2017-11-01T00:00:00Z date_updated: 2021-01-12T08:06:22Z day: '01' department: - _id: LaEr doi: 10.1214/16-AIHP765 ec_funded: 1 intvolume: ' 53' issue: '4' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1504.00650 month: '11' oa: 1 oa_version: Submitted Version page: 1606 - 1656 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Annales de l'institut Henri Poincare (B) Probability and Statistics publication_identifier: issn: - '02460203' publication_status: published publisher: Institute of Mathematical Statistics publist_id: '7189' quality_controlled: '1' scopus_import: 1 status: public title: Universality for random matrix flows with time dependent density type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 53 year: '2017' ... --- _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: '550' abstract: - lang: eng text: For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities. article_number: '63' author: - first_name: Johannes full_name: Alt, Johannes id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87 last_name: Alt citation: ama: Alt J. Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. 2017;22. doi:10.1214/17-ECP97 apa: Alt, J. (2017). Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/17-ECP97 chicago: Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” Electronic Communications in Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/17-ECP97. ieee: J. Alt, “Singularities of the density of states of random Gram matrices,” Electronic Communications in Probability, vol. 22. Institute of Mathematical Statistics, 2017. ista: Alt J. 2017. Singularities of the density of states of random Gram matrices. Electronic Communications in Probability. 22, 63. mla: Alt, Johannes. “Singularities of the Density of States of Random Gram Matrices.” Electronic Communications in Probability, vol. 22, 63, Institute of Mathematical Statistics, 2017, doi:10.1214/17-ECP97. short: J. Alt, Electronic Communications in Probability 22 (2017). date_created: 2018-12-11T11:47:07Z date_published: 2017-11-21T00:00:00Z date_updated: 2023-09-07T12:38:08Z day: '21' ddc: - '539' department: - _id: LaEr doi: 10.1214/17-ECP97 ec_funded: 1 file: - access_level: open_access checksum: 0ec05303a0de190de145654237984c79 content_type: application/pdf creator: system date_created: 2018-12-12T10:08:04Z date_updated: 2020-07-14T12:47:00Z file_id: '4663' file_name: IST-2018-926-v1+1_euclid.ecp.1511233247.pdf file_size: 470876 relation: main_file file_date_updated: 2020-07-14T12:47:00Z has_accepted_license: '1' intvolume: ' 22' language: - iso: eng month: '11' 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: Electronic Communications in Probability publication_identifier: issn: - 1083589X publication_status: published publisher: Institute of Mathematical Statistics publist_id: '7265' pubrep_id: '926' quality_controlled: '1' related_material: record: - id: '149' relation: dissertation_contains status: public scopus_import: 1 status: public title: Singularities of the density of states of random Gram 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 22 year: '2017' ... --- _id: '1144' abstract: - lang: eng text: We show that matrix elements of functions of N × N Wigner matrices fluctuate on a scale of order N−1/2 and we identify the limiting fluctuation. Our result holds for any function f of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11]. acknowledgement: Partially supported by the IST Austria Excellence Scholarship. article_number: '86' author: - 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: Erdös L, Schröder DJ. Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. 2017;21. doi:10.1214/16-ECP38 apa: Erdös, L., & Schröder, D. J. (2017). Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/16-ECP38 chicago: Erdös, László, and Dominik J Schröder. “Fluctuations of Functions of Wigner Matrices.” Electronic Communications in Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/16-ECP38. ieee: L. Erdös and D. J. Schröder, “Fluctuations of functions of Wigner matrices,” Electronic Communications in Probability, vol. 21. Institute of Mathematical Statistics, 2017. ista: Erdös L, Schröder DJ. 2017. Fluctuations of functions of Wigner matrices. Electronic Communications in Probability. 21, 86. mla: Erdös, László, and Dominik J. Schröder. “Fluctuations of Functions of Wigner Matrices.” Electronic Communications in Probability, vol. 21, 86, Institute of Mathematical Statistics, 2017, doi:10.1214/16-ECP38. short: L. Erdös, D.J. Schröder, Electronic Communications in Probability 21 (2017). date_created: 2018-12-11T11:50:23Z date_published: 2017-01-02T00:00:00Z date_updated: 2023-09-07T12:54:12Z day: '02' ddc: - '510' department: - _id: LaEr doi: 10.1214/16-ECP38 ec_funded: 1 file: - access_level: open_access content_type: application/pdf creator: system date_created: 2018-12-12T10:18:10Z date_updated: 2018-12-12T10:18:10Z file_id: '5329' file_name: IST-2017-747-v1+1_euclid.ecp.1483347665.pdf file_size: 440770 relation: main_file file_date_updated: 2018-12-12T10:18:10Z has_accepted_license: '1' intvolume: ' 21' language: - iso: eng month: '01' 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: Electronic Communications in Probability publication_status: published publisher: Institute of Mathematical Statistics publist_id: '6214' pubrep_id: '747' quality_controlled: '1' related_material: record: - id: '6179' relation: dissertation_contains status: public scopus_import: 1 status: public title: Fluctuations of functions of 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 21 year: '2017' ... --- _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: '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: '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: '1023' abstract: - lang: eng text: We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε. article_number: '22' article_processing_charge: No author: - first_name: Yuriy full_name: Nemish, Yuriy id: 4D902E6A-F248-11E8-B48F-1D18A9856A87 last_name: Nemish orcid: 0000-0002-7327-856X citation: ama: Nemish Y. Local law for the product of independent non-Hermitian random matrices with independent entries. Electronic Journal of Probability. 2017;22. doi:10.1214/17-EJP38 apa: Nemish, Y. (2017). Local law for the product of independent non-Hermitian random matrices with independent entries. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/17-EJP38 chicago: Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random Matrices with Independent Entries.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/17-EJP38. ieee: Y. Nemish, “Local law for the product of independent non-Hermitian random matrices with independent entries,” Electronic Journal of Probability, vol. 22. Institute of Mathematical Statistics, 2017. ista: Nemish Y. 2017. Local law for the product of independent non-Hermitian random matrices with independent entries. Electronic Journal of Probability. 22, 22. mla: Nemish, Yuriy. “Local Law for the Product of Independent Non-Hermitian Random Matrices with Independent Entries.” Electronic Journal of Probability, vol. 22, 22, Institute of Mathematical Statistics, 2017, doi:10.1214/17-EJP38. short: Y. Nemish, Electronic Journal of Probability 22 (2017). date_created: 2018-12-11T11:49:44Z date_published: 2017-02-06T00:00:00Z date_updated: 2023-09-22T09:27:51Z day: '06' ddc: - '510' department: - _id: LaEr doi: 10.1214/17-EJP38 external_id: isi: - '000396611900022' file: - access_level: open_access content_type: application/pdf creator: system date_created: 2018-12-12T10:15:29Z date_updated: 2018-12-12T10:15:29Z file_id: '5149' file_name: IST-2017-802-v1+1_euclid.ejp.1487991681.pdf file_size: 742275 relation: main_file file_date_updated: 2018-12-12T10:15:29Z has_accepted_license: '1' intvolume: ' 22' isi: 1 language: - iso: eng month: '02' oa: 1 oa_version: Published Version publication: Electronic Journal of Probability publication_identifier: issn: - '10836489' publication_status: published publisher: Institute of Mathematical Statistics publist_id: '6370' pubrep_id: '802' quality_controlled: '1' scopus_import: '1' status: public title: Local law for the product of independent non-Hermitian random matrices with independent 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: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 22 year: '2017' ... --- _id: '1010' abstract: - lang: eng text: 'We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. ' article_number: '25' article_processing_charge: No author: - first_name: Johannes full_name: Alt, Johannes id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87 last_name: Alt - 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: Alt J, Erdös L, Krüger TH. Local law for random Gram matrices. Electronic Journal of Probability. 2017;22. doi:10.1214/17-EJP42 apa: Alt, J., Erdös, L., & Krüger, T. H. (2017). Local law for random Gram matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/17-EJP42 chicago: Alt, Johannes, László Erdös, and Torben H Krüger. “Local Law for Random Gram Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2017. https://doi.org/10.1214/17-EJP42. ieee: J. Alt, L. Erdös, and T. H. Krüger, “Local law for random Gram matrices,” Electronic Journal of Probability, vol. 22. Institute of Mathematical Statistics, 2017. ista: Alt J, Erdös L, Krüger TH. 2017. Local law for random Gram matrices. Electronic Journal of Probability. 22, 25. mla: Alt, Johannes, et al. “Local Law for Random Gram Matrices.” Electronic Journal of Probability, vol. 22, 25, Institute of Mathematical Statistics, 2017, doi:10.1214/17-EJP42. short: J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017). date_created: 2018-12-11T11:49:40Z date_published: 2017-03-08T00:00:00Z date_updated: 2023-09-22T09:45:23Z day: '08' ddc: - '510' - '539' department: - _id: LaEr doi: 10.1214/17-EJP42 ec_funded: 1 external_id: arxiv: - '1606.07353' isi: - '000396611900025' file: - access_level: open_access content_type: application/pdf creator: system date_created: 2018-12-12T10:13:39Z date_updated: 2018-12-12T10:13:39Z file_id: '5024' file_name: IST-2017-807-v1+1_euclid.ejp.1488942016.pdf file_size: 639384 relation: main_file file_date_updated: 2018-12-12T10:13:39Z has_accepted_license: '1' intvolume: ' 22' isi: 1 language: - iso: eng month: '03' 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: Electronic Journal of Probability publication_identifier: issn: - '10836489' publication_status: published publisher: Institute of Mathematical Statistics publist_id: '6386' pubrep_id: '807' quality_controlled: '1' related_material: record: - id: '149' relation: dissertation_contains status: public scopus_import: '1' status: public title: Local law for random Gram 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: 22 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: '447' abstract: - lang: eng text: We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015). article_processing_charge: No article_type: original author: - first_name: Patrik full_name: Ferrari, Patrik last_name: Ferrari - first_name: Peter full_name: Nejjar, Peter id: 4BF426E2-F248-11E8-B48F-1D18A9856A87 last_name: Nejjar citation: ama: Ferrari P, Nejjar P. Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. 2017;9:299-325. doi:10.30757/ALEA.v14-17 apa: Ferrari, P., & Nejjar, P. (2017). Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. Instituto Nacional de Matematica Pura e Aplicada. https://doi.org/10.30757/ALEA.v14-17 chicago: Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface in Presence of Shocks.” Revista Latino-Americana de Probabilidade e Estatística. Instituto Nacional de Matematica Pura e Aplicada, 2017. https://doi.org/10.30757/ALEA.v14-17. ieee: P. Ferrari and P. Nejjar, “Fluctuations of the competition interface in presence of shocks,” Revista Latino-Americana de Probabilidade e Estatística, vol. 9. Instituto Nacional de Matematica Pura e Aplicada, pp. 299–325, 2017. ista: Ferrari P, Nejjar P. 2017. Fluctuations of the competition interface in presence of shocks. Revista Latino-Americana de Probabilidade e Estatística. 9, 299–325. mla: Ferrari, Patrik, and Peter Nejjar. “Fluctuations of the Competition Interface in Presence of Shocks.” Revista Latino-Americana de Probabilidade e Estatística, vol. 9, Instituto Nacional de Matematica Pura e Aplicada, 2017, pp. 299–325, doi:10.30757/ALEA.v14-17. short: P. Ferrari, P. Nejjar, Revista Latino-Americana de Probabilidade e Estatística 9 (2017) 299–325. date_created: 2018-12-11T11:46:31Z date_published: 2017-03-23T00:00:00Z date_updated: 2023-10-10T13:10:32Z day: '23' department: - _id: LaEr - _id: JaMa doi: 10.30757/ALEA.v14-17 ec_funded: 1 intvolume: ' 9' language: - iso: eng main_file_link: - open_access: '1' url: http://alea.impa.br/articles/v14/14-17.pdf month: '03' oa: 1 oa_version: Submitted Version page: 299 - 325 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Revista Latino-Americana de Probabilidade e Estatística publication_status: published publisher: Instituto Nacional de Matematica Pura e Aplicada publist_id: '7376' quality_controlled: '1' scopus_import: '1' status: public title: Fluctuations of the competition interface in presence of shocks type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 9 year: '2017' ... --- _id: '1157' abstract: - lang: eng text: We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that σ is diagonal and that the entries of X have a sub-exponential decay. acknowledgement: "We thank Horng-Tzer Yau for numerous discussions and remarks. We are grateful to Ben Adlam, Jinho Baik, Zhigang Bao, Paul Bourgade, László Erd ̋os, Iain Johnstone and Antti Knowles for comments. We are also grate-\r\nful to the anonymous referee for carefully reading our manuscript and suggesting several improvements." author: - first_name: Ji full_name: Lee, Ji last_name: Lee - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 2016;26(6):3786-3839. doi:10.1214/16-AAP1193 apa: Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AAP1193 chicago: Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/16-AAP1193. ieee: J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” Annals of Applied Probability, vol. 26, no. 6. Institute of Mathematical Statistics, pp. 3786–3839, 2016. ista: Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 26(6), 3786–3839. mla: Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:10.1214/16-AAP1193. short: J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839. date_created: 2018-12-11T11:50:27Z date_published: 2016-12-15T00:00:00Z date_updated: 2021-01-12T06:48:43Z day: '15' department: - _id: LaEr doi: 10.1214/16-AAP1193 ec_funded: 1 intvolume: ' 26' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1409.4979 month: '12' oa: 1 oa_version: Preprint page: 3786 - 3839 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Annals of Applied Probability publication_status: published publisher: Institute of Mathematical Statistics publist_id: '6201' quality_controlled: '1' scopus_import: 1 status: public title: Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 26 year: '2016' ... --- _id: '1219' abstract: - lang: eng text: We consider N×N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues ofW and V are typically of the same order. For a large class of diagonal matrices V , we show that the local statistics in the bulk of the spectrum are universal in the limit of large N. acknowledgement: "J.C. was supported in part by National Research Foundation of Korea Grant 2011-0013474 and TJ Park Junior Faculty Fellowship.\r\nK.S. was supported by ERC Advanced Grant RANMAT, No. 338804, and the \"Fund for Math.\"\r\nB.S. was supported by NSF GRFP Fellowship DGE-1144152.\r\nH.Y. was supported in part by NSF Grant DMS-13-07444 and Simons investigator fellowship. We thank Paul Bourgade, László Erd ̋os and Antti Knowles for helpful comments. We are grateful to the Taida Institute for Mathematical\r\nSciences and National Taiwan Universality for their hospitality during part of this\r\nresearch. We thank Thomas Spencer and the Institute for Advanced Study for their\r\nhospitality during the academic year 2013–2014. " author: - first_name: Jioon full_name: Lee, Jioon last_name: Lee - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 - first_name: Ben full_name: Stetler, Ben last_name: Stetler - first_name: Horngtzer full_name: Yau, Horngtzer last_name: Yau citation: ama: Lee J, Schnelli K, Stetler B, Yau H. Bulk universality for deformed wigner matrices. Annals of Probability. 2016;44(3):2349-2425. doi:10.1214/15-AOP1023 apa: Lee, J., Schnelli, K., Stetler, B., & Yau, H. (2016). Bulk universality for deformed wigner matrices. Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/15-AOP1023 chicago: Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality for Deformed Wigner Matrices.” Annals of Probability. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/15-AOP1023. ieee: J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed wigner matrices,” Annals of Probability, vol. 44, no. 3. Institute of Mathematical Statistics, pp. 2349–2425, 2016. ista: Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed wigner matrices. Annals of Probability. 44(3), 2349–2425. mla: Lee, Jioon, et al. “Bulk Universality for Deformed Wigner Matrices.” Annals of Probability, vol. 44, no. 3, Institute of Mathematical Statistics, 2016, pp. 2349–425, doi:10.1214/15-AOP1023. short: J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425. date_created: 2018-12-11T11:50:47Z date_published: 2016-01-01T00:00:00Z date_updated: 2021-01-12T06:49:10Z day: '01' department: - _id: LaEr doi: 10.1214/15-AOP1023 ec_funded: 1 intvolume: ' 44' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1405.6634 month: '01' oa: 1 oa_version: Preprint page: 2349 - 2425 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_status: published publisher: Institute of Mathematical Statistics publist_id: '6115' quality_controlled: '1' scopus_import: 1 status: public title: Bulk universality for deformed wigner matrices type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 44 year: '2016' ... --- _id: '1223' abstract: - lang: eng text: We consider a random Schrödinger operator on the binary tree with a random potential which is the sum of a random radially symmetric potential, Qr, and a random transversally periodic potential, κQt, with coupling constant κ. Using a new one-dimensional dynamical systems approach combined with Jensen's inequality in hyperbolic space (our key estimate) we obtain a fractional moment estimate proving localization for small and large κ. Together with a previous result we therefore obtain a model with two Anderson transitions, from localization to delocalization and back to localization, when increasing κ. As a by-product we also have a partially new proof of one-dimensional Anderson localization at any disorder. author: - first_name: Richard full_name: Froese, Richard last_name: Froese - first_name: Darrick full_name: Lee, Darrick last_name: Lee - first_name: Christian full_name: Sadel, Christian id: 4760E9F8-F248-11E8-B48F-1D18A9856A87 last_name: Sadel orcid: 0000-0001-8255-3968 - first_name: Wolfgang full_name: Spitzer, Wolfgang last_name: Spitzer - first_name: Günter full_name: Stolz, Günter last_name: Stolz citation: ama: Froese R, Lee D, Sadel C, Spitzer W, Stolz G. Localization for transversally periodic random potentials on binary trees. Journal of Spectral Theory. 2016;6(3):557-600. doi:10.4171/JST/132 apa: Froese, R., Lee, D., Sadel, C., Spitzer, W., & Stolz, G. (2016). Localization for transversally periodic random potentials on binary trees. Journal of Spectral Theory. European Mathematical Society. https://doi.org/10.4171/JST/132 chicago: Froese, Richard, Darrick Lee, Christian Sadel, Wolfgang Spitzer, and Günter Stolz. “Localization for Transversally Periodic Random Potentials on Binary Trees.” Journal of Spectral Theory. European Mathematical Society, 2016. https://doi.org/10.4171/JST/132. ieee: R. Froese, D. Lee, C. Sadel, W. Spitzer, and G. Stolz, “Localization for transversally periodic random potentials on binary trees,” Journal of Spectral Theory, vol. 6, no. 3. European Mathematical Society, pp. 557–600, 2016. ista: Froese R, Lee D, Sadel C, Spitzer W, Stolz G. 2016. Localization for transversally periodic random potentials on binary trees. Journal of Spectral Theory. 6(3), 557–600. mla: Froese, Richard, et al. “Localization for Transversally Periodic Random Potentials on Binary Trees.” Journal of Spectral Theory, vol. 6, no. 3, European Mathematical Society, 2016, pp. 557–600, doi:10.4171/JST/132. short: R. Froese, D. Lee, C. Sadel, W. Spitzer, G. Stolz, Journal of Spectral Theory 6 (2016) 557–600. date_created: 2018-12-11T11:50:48Z date_published: 2016-01-01T00:00:00Z date_updated: 2021-01-12T06:49:12Z day: '01' department: - _id: LaEr doi: 10.4171/JST/132 intvolume: ' 6' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1408.3961 month: '01' oa: 1 oa_version: Preprint page: 557 - 600 publication: Journal of Spectral Theory publication_status: published publisher: European Mathematical Society publist_id: '6112' quality_controlled: '1' scopus_import: 1 status: public title: Localization for transversally periodic random potentials on binary trees type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 6 year: '2016' ... --- _id: '1257' abstract: - lang: eng text: We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix (Formula presented.). Focusing on the eigenvalues of (Formula presented.) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required (Formula presented.) to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work. acknowledgement: Open access funding provided by Institute of Science and Technology (IST Austria). The work of C. Sadel was supported by NSERC Discovery Grant 92997-2010 RGPIN and by the People Programme (Marie Curie Actions) of the EU 7th Framework Programme FP7/2007-2013, REA Grant 291734. article_processing_charge: Yes (via OA deal) author: - first_name: Christian full_name: Sadel, Christian id: 4760E9F8-F248-11E8-B48F-1D18A9856A87 last_name: Sadel orcid: 0000-0001-8255-3968 - first_name: Bálint full_name: Virág, Bálint last_name: Virág citation: ama: Sadel C, Virág B. A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. 2016;343(3):881-919. doi:10.1007/s00220-016-2600-4 apa: Sadel, C., & Virág, B. (2016). A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-016-2600-4 chicago: Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.” Communications in Mathematical Physics. Springer, 2016. https://doi.org/10.1007/s00220-016-2600-4. ieee: C. Sadel and B. Virág, “A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes,” Communications in Mathematical Physics, vol. 343, no. 3. Springer, pp. 881–919, 2016. ista: Sadel C, Virág B. 2016. A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. 343(3), 881–919. mla: Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.” Communications in Mathematical Physics, vol. 343, no. 3, Springer, 2016, pp. 881–919, doi:10.1007/s00220-016-2600-4. short: C. Sadel, B. Virág, Communications in Mathematical Physics 343 (2016) 881–919. date_created: 2018-12-11T11:50:59Z date_published: 2016-05-01T00:00:00Z date_updated: 2021-01-12T06:49:26Z day: '01' ddc: - '510' - '539' department: - _id: LaEr doi: 10.1007/s00220-016-2600-4 ec_funded: 1 file: - access_level: open_access checksum: 4fb2411d9c2f56676123165aad46c828 content_type: application/pdf creator: system date_created: 2018-12-12T10:15:02Z date_updated: 2020-07-14T12:44:42Z file_id: '5119' file_name: IST-2016-703-v1+1_s00220-016-2600-4.pdf file_size: 800792 relation: main_file file_date_updated: 2020-07-14T12:44:42Z has_accepted_license: '1' intvolume: ' 343' issue: '3' language: - iso: eng month: '05' oa: 1 oa_version: Published Version page: 881 - 919 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Communications in Mathematical Physics publication_status: published publisher: Springer publist_id: '6067' pubrep_id: '703' quality_controlled: '1' scopus_import: 1 status: public title: A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes 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: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 343 year: '2016' ... --- _id: '1280' abstract: - lang: eng text: We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics. acknowledgement: "The work of P.B. was partially supported by National Sci-\r\nence Foundation Grant DMS-1208859. The work of L.E. was partially supported\r\nby ERC Advanced Grant RANMAT 338804. The work of H.-T. Y. was partially\r\nsupported by National Science Foundation Grant DMS-1307444 and a Simons In-\r\nvestigator award. \ The work of J.Y. was partially supported by National Science\r\nFoundation Grant DMS-1207961. The major part of this research was conducted\r\nwhen all authors were visiting IAS and were also supported by National Science\r\nFoundation Grant DMS-1128255." author: - first_name: Paul full_name: Bourgade, Paul last_name: Bourgade - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Horngtzer full_name: Yau, Horngtzer last_name: Yau - first_name: Jun full_name: Yin, Jun last_name: Yin citation: ama: Bourgade P, Erdös L, Yau H, Yin J. Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. 2016;69(10):1815-1881. doi:10.1002/cpa.21624 apa: Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2016). Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. Wiley-Blackwell. https://doi.org/10.1002/cpa.21624 chicago: Bourgade, Paul, László Erdös, Horngtzer Yau, and Jun Yin. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics. Wiley-Blackwell, 2016. https://doi.org/10.1002/cpa.21624. ieee: P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Fixed energy universality for generalized wigner matrices,” Communications on Pure and Applied Mathematics, vol. 69, no. 10. Wiley-Blackwell, pp. 1815–1881, 2016. ista: Bourgade P, Erdös L, Yau H, Yin J. 2016. Fixed energy universality for generalized wigner matrices. Communications on Pure and Applied Mathematics. 69(10), 1815–1881. mla: Bourgade, Paul, et al. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics, vol. 69, no. 10, Wiley-Blackwell, 2016, pp. 1815–81, doi:10.1002/cpa.21624. short: P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied Mathematics 69 (2016) 1815–1881. date_created: 2018-12-11T11:51:07Z date_published: 2016-10-01T00:00:00Z date_updated: 2021-01-12T06:49:35Z day: '01' department: - _id: LaEr doi: 10.1002/cpa.21624 ec_funded: 1 intvolume: ' 69' issue: '10' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1407.5606 month: '10' oa: 1 oa_version: Preprint page: 1815 - 1881 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_status: published publisher: Wiley-Blackwell publist_id: '6036' scopus_import: 1 status: public title: Fixed energy universality for generalized wigner matrices type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 69 year: '2016' ... --- _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: '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: '1608' abstract: - lang: eng text: 'We show that the Anderson model has a transition from localization to delocalization at exactly 2 dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At d dimensional growth for d>2 this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform d dimensional growth with d<2 one has pure point spectrum in this energy region. At exactly uniform 2 dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum (d≤2) to absolutely continuous spectrum (d≥3) for random operators of the type rΔdr+λ on ℤd, where r is an orthogonal radial projection, Δd the discrete adjacency operator (Laplacian) on ℤd and λ a random potential. ' author: - first_name: Christian full_name: Sadel, Christian id: 4760E9F8-F248-11E8-B48F-1D18A9856A87 last_name: Sadel orcid: 0000-0001-8255-3968 citation: ama: Sadel C. Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Annales Henri Poincare. 2016;17(7):1631-1675. doi:10.1007/s00023-015-0456-3 apa: Sadel, C. (2016). Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Annales Henri Poincare. Birkhäuser. https://doi.org/10.1007/s00023-015-0456-3 chicago: Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel.” Annales Henri Poincare. Birkhäuser, 2016. https://doi.org/10.1007/s00023-015-0456-3. ieee: C. Sadel, “Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel,” Annales Henri Poincare, vol. 17, no. 7. Birkhäuser, pp. 1631–1675, 2016. ista: Sadel C. 2016. Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Annales Henri Poincare. 17(7), 1631–1675. mla: Sadel, Christian. “Anderson Transition at 2 Dimensional Growth Rate on Antitrees and Spectral Theory for Operators with One Propagating Channel.” Annales Henri Poincare, vol. 17, no. 7, Birkhäuser, 2016, pp. 1631–75, doi:10.1007/s00023-015-0456-3. short: C. Sadel, Annales Henri Poincare 17 (2016) 1631–1675. date_created: 2018-12-11T11:53:00Z date_published: 2016-07-01T00:00:00Z date_updated: 2021-01-12T06:51:58Z day: '01' department: - _id: LaEr doi: 10.1007/s00023-015-0456-3 ec_funded: 1 intvolume: ' 17' issue: '7' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1501.04287 month: '07' oa: 1 oa_version: Preprint page: 1631 - 1675 project: - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: Annales Henri Poincare publication_status: published publisher: Birkhäuser publist_id: '5558' quality_controlled: '1' scopus_import: 1 status: public title: Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 17 year: '2016' ... --- _id: '1881' abstract: - lang: eng text: 'We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues. ' acknowledgement: "Most of the presented work was obtained while Kevin Schnelli was staying at the IAS with the support of\r\nThe Fund For Math." author: - first_name: Jioon full_name: Lee, Jioon last_name: Lee - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 2016;164(1-2):165-241. doi:10.1007/s00440-014-0610-8 apa: Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-014-0610-8 chicago: Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields. Springer, 2016. https://doi.org/10.1007/s00440-014-0610-8. ieee: J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed Wigner matrices,” Probability Theory and Related Fields, vol. 164, no. 1–2. Springer, pp. 165–241, 2016. ista: Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241. mla: Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices.” Probability Theory and Related Fields, vol. 164, no. 1–2, Springer, 2016, pp. 165–241, doi:10.1007/s00440-014-0610-8. short: J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241. date_created: 2018-12-11T11:54:31Z date_published: 2016-02-01T00:00:00Z date_updated: 2021-01-12T06:53:49Z day: '01' department: - _id: LaEr doi: 10.1007/s00440-014-0610-8 ec_funded: 1 intvolume: ' 164' issue: 1-2 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1310.7057 month: '02' oa: 1 oa_version: Preprint page: 165 - 241 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_status: published publisher: Springer publist_id: '5215' quality_controlled: '1' scopus_import: 1 status: public title: Extremal eigenvalues and eigenvectors of deformed Wigner matrices type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 164 year: '2016' ... --- _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: '1508' abstract: - lang: eng text: We consider generalized Wigner ensembles and general β-ensembles with analytic potentials for any β ≥ 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian β-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any C4(ℝ) potential. author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Horng full_name: Yau, Horng last_name: Yau citation: ama: Erdös L, Yau H. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 2015;17(8):1927-2036. doi:10.4171/JEMS/548 apa: Erdös, L., & Yau, H. (2015). Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. European Mathematical Society. https://doi.org/10.4171/JEMS/548 chicago: Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” Journal of the European Mathematical Society. European Mathematical Society, 2015. https://doi.org/10.4171/JEMS/548. ieee: L. Erdös and H. Yau, “Gap universality of generalized Wigner and β ensembles,” Journal of the European Mathematical Society, vol. 17, no. 8. European Mathematical Society, pp. 1927–2036, 2015. ista: Erdös L, Yau H. 2015. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 17(8), 1927–2036. mla: Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” Journal of the European Mathematical Society, vol. 17, no. 8, European Mathematical Society, 2015, pp. 1927–2036, doi:10.4171/JEMS/548. short: L. Erdös, H. Yau, Journal of the European Mathematical Society 17 (2015) 1927–2036. date_created: 2018-12-11T11:52:26Z date_published: 2015-08-01T00:00:00Z date_updated: 2021-01-12T06:51:15Z day: '01' department: - _id: LaEr doi: 10.4171/JEMS/548 intvolume: ' 17' issue: '8' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1211.3786 month: '08' oa: 1 oa_version: Preprint page: 1927 - 2036 publication: Journal of the European Mathematical Society publication_status: published publisher: European Mathematical Society publist_id: '5669' quality_controlled: '1' scopus_import: 1 status: public title: Gap universality of generalized Wigner and β ensembles type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 17 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' ... --- _id: '1674' abstract: - lang: eng text: We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix. article_number: '1550018' author: - first_name: Jioon full_name: Lee, Jioon last_name: Lee - first_name: Kevin full_name: Schnelli, Kevin id: 434AD0AE-F248-11E8-B48F-1D18A9856A87 last_name: Schnelli orcid: 0000-0003-0954-3231 citation: ama: Lee J, Schnelli K. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 2015;27(8). doi:10.1142/S0129055X1550018X apa: Lee, J., & Schnelli, K. (2015). Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. World Scientific Publishing. https://doi.org/10.1142/S0129055X1550018X chicago: Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics. World Scientific Publishing, 2015. https://doi.org/10.1142/S0129055X1550018X. ieee: J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” Reviews in Mathematical Physics, vol. 27, no. 8. World Scientific Publishing, 2015. ista: Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 27(8), 1550018. mla: Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics, vol. 27, no. 8, 1550018, World Scientific Publishing, 2015, doi:10.1142/S0129055X1550018X. short: J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015). date_created: 2018-12-11T11:53:24Z date_published: 2015-09-01T00:00:00Z date_updated: 2021-01-12T06:52:26Z day: '01' department: - _id: LaEr doi: 10.1142/S0129055X1550018X intvolume: ' 27' issue: '8' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1407.8015 month: '09' oa: 1 oa_version: Preprint publication: Reviews in Mathematical Physics publication_status: published publisher: World Scientific Publishing publist_id: '5475' quality_controlled: '1' scopus_import: 1 status: public title: Edge universality for deformed Wigner matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 27 year: '2015' ... --- _id: '1824' abstract: - lang: eng text: Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates. article_number: '6977' author: - first_name: Johannes full_name: Knebel, Johannes last_name: Knebel - first_name: Markus full_name: Weber, Markus last_name: Weber - 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: Erwin full_name: Frey, Erwin last_name: Frey citation: ama: Knebel J, Weber M, Krüger TH, Frey E. Evolutionary games of condensates in coupled birth-death processes. Nature Communications. 2015;6. doi:10.1038/ncomms7977 apa: Knebel, J., Weber, M., Krüger, T. H., & Frey, E. (2015). Evolutionary games of condensates in coupled birth-death processes. Nature Communications. Nature Publishing Group. https://doi.org/10.1038/ncomms7977 chicago: Knebel, Johannes, Markus Weber, Torben H Krüger, and Erwin Frey. “Evolutionary Games of Condensates in Coupled Birth-Death Processes.” Nature Communications. Nature Publishing Group, 2015. https://doi.org/10.1038/ncomms7977. ieee: J. Knebel, M. Weber, T. H. Krüger, and E. Frey, “Evolutionary games of condensates in coupled birth-death processes,” Nature Communications, vol. 6. Nature Publishing Group, 2015. ista: Knebel J, Weber M, Krüger TH, Frey E. 2015. Evolutionary games of condensates in coupled birth-death processes. Nature Communications. 6, 6977. mla: Knebel, Johannes, et al. “Evolutionary Games of Condensates in Coupled Birth-Death Processes.” Nature Communications, vol. 6, 6977, Nature Publishing Group, 2015, doi:10.1038/ncomms7977. short: J. Knebel, M. Weber, T.H. Krüger, E. Frey, Nature Communications 6 (2015). date_created: 2018-12-11T11:54:13Z date_published: 2015-04-24T00:00:00Z date_updated: 2021-01-12T06:53:26Z day: '24' ddc: - '530' department: - _id: LaEr doi: 10.1038/ncomms7977 file: - access_level: open_access checksum: c4cffb5c8b245e658a34eac71a03e7cc content_type: application/pdf creator: system date_created: 2018-12-12T10:16:54Z date_updated: 2020-07-14T12:45:17Z file_id: '5245' file_name: IST-2016-451-v1+1_ncomms7977.pdf file_size: 1151501 relation: main_file file_date_updated: 2020-07-14T12:45:17Z has_accepted_license: '1' intvolume: ' 6' language: - iso: eng month: '04' oa: 1 oa_version: Published Version publication: Nature Communications publication_status: published publisher: Nature Publishing Group publist_id: '5282' pubrep_id: '451' quality_controlled: '1' scopus_import: 1 status: public title: Evolutionary games of condensates in coupled birth-death processes 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: 6 year: '2015' ... --- _id: '1864' abstract: - lang: eng text: "The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas.\r\n" author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Antti full_name: Knowles, Antti last_name: Knowles citation: ama: 'Erdös L, Knowles A. The Altshuler–Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 2015;16(3):709-799. doi:10.1007/s00023-014-0333-5' apa: 'Erdös, L., & Knowles, A. (2015). The Altshuler–Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. Springer. https://doi.org/10.1007/s00023-014-0333-5' chicago: 'Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random Band Matrices II: The General Case.” Annales Henri Poincare. Springer, 2015. https://doi.org/10.1007/s00023-014-0333-5.' ieee: 'L. Erdös and A. Knowles, “The Altshuler–Shklovskii formulas for random band matrices II: The general case,” Annales Henri Poincare, vol. 16, no. 3. Springer, pp. 709–799, 2015.' ista: 'Erdös L, Knowles A. 2015. The Altshuler–Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 16(3), 709–799.' mla: 'Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random Band Matrices II: The General Case.” Annales Henri Poincare, vol. 16, no. 3, Springer, 2015, pp. 709–99, doi:10.1007/s00023-014-0333-5.' short: L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799. date_created: 2018-12-11T11:54:26Z date_published: 2015-03-01T00:00:00Z date_updated: 2021-01-12T06:53:42Z day: '01' department: - _id: LaEr doi: 10.1007/s00023-014-0333-5 ec_funded: 1 intvolume: ' 16' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1309.5107 month: '03' oa: 1 oa_version: Preprint page: 709 - 799 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Annales Henri Poincare publication_status: published publisher: Springer publist_id: '5233' scopus_import: 1 status: public title: 'The Altshuler–Shklovskii formulas for random band matrices II: The general case' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 16 year: '2015' ... --- _id: '2166' abstract: - lang: eng text: 'We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson- Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it describes the correlations of the eigenvalue density in general metallic sampleswith weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev Lett 74:2563-2566, 1995). The proof is given in the current paper and its companion (Ann. H. Poincaré. arXiv:1309.5107, 2014). ' author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Antti full_name: Knowles, Antti last_name: Knowles citation: ama: 'Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 2015;333(3):1365-1416. doi:10.1007/s00220-014-2119-5' apa: 'Erdös, L., & Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2119-5' chicago: 'Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics. Springer, 2015. https://doi.org/10.1007/s00220-014-2119-5.' ieee: 'L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case,” Communications in Mathematical Physics, vol. 333, no. 3. Springer, pp. 1365–1416, 2015.' ista: 'Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 333(3), 1365–1416.' mla: 'Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics, vol. 333, no. 3, Springer, 2015, pp. 1365–416, doi:10.1007/s00220-014-2119-5.' short: L. Erdös, A. Knowles, Communications in Mathematical Physics 333 (2015) 1365–1416. date_created: 2018-12-11T11:56:05Z date_published: 2015-02-01T00:00:00Z date_updated: 2021-01-12T06:55:43Z day: '01' department: - _id: LaEr doi: 10.1007/s00220-014-2119-5 intvolume: ' 333' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1309.5106 month: '02' oa: 1 oa_version: Preprint page: 1365 - 1416 publication: Communications in Mathematical Physics publication_status: published publisher: Springer publist_id: '4818' quality_controlled: '1' scopus_import: 1 status: public title: 'The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case' type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 333 year: '2015' ... --- _id: '1677' abstract: - lang: eng text: We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN-y,N-x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. Italso occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale. article_number: '103301' author: - first_name: Johannes full_name: Alt, Johannes id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87 last_name: Alt citation: ama: Alt J. The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. 2015;56(10). doi:10.1063/1.4932606 apa: Alt, J. (2015). The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. American Institute of Physics. https://doi.org/10.1063/1.4932606 chicago: Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” Journal of Mathematical Physics. American Institute of Physics, 2015. https://doi.org/10.1063/1.4932606. ieee: J. Alt, “The local semicircle law for random matrices with a fourfold symmetry,” Journal of Mathematical Physics, vol. 56, no. 10. American Institute of Physics, 2015. ista: Alt J. 2015. The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. 56(10), 103301. mla: Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” Journal of Mathematical Physics, vol. 56, no. 10, 103301, American Institute of Physics, 2015, doi:10.1063/1.4932606. short: J. Alt, Journal of Mathematical Physics 56 (2015). date_created: 2018-12-11T11:53:25Z date_published: 2015-10-09T00:00:00Z date_updated: 2023-09-07T12:38:08Z day: '09' department: - _id: LaEr doi: 10.1063/1.4932606 ec_funded: 1 intvolume: ' 56' issue: '10' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1506.04683 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 Mathematical Physics publication_status: published publisher: American Institute of Physics publist_id: '5472' quality_controlled: '1' related_material: record: - id: '149' relation: dissertation_contains status: public scopus_import: 1 status: public title: The local semicircle law for random matrices with a fourfold symmetry type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 56 year: '2015' ... --- _id: '1926' abstract: - lang: eng text: We consider cross products of finite graphs with a class of trees that have arbitrarily but finitely long line segments, such as the Fibonacci tree. Such cross products are called tree-strips. We prove that for small disorder random Schrödinger operators on such tree-strips have purely absolutely continuous spectrum in a certain set. article_processing_charge: No article_type: original author: - first_name: Christian full_name: Sadel, Christian id: 4760E9F8-F248-11E8-B48F-1D18A9856A87 last_name: Sadel orcid: 0000-0001-8255-3968 citation: ama: Sadel C. Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry. 2014;17(3-4):409-440. doi:10.1007/s11040-014-9163-4 apa: Sadel, C. (2014). Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry. Springer. https://doi.org/10.1007/s11040-014-9163-4 chicago: Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators on the Fibonacci and Similar Tree-Strips.” Mathematical Physics, Analysis and Geometry. Springer, 2014. https://doi.org/10.1007/s11040-014-9163-4. ieee: C. Sadel, “Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips,” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4. Springer, pp. 409–440, 2014. ista: Sadel C. 2014. Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry. 17(3–4), 409–440. mla: Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators on the Fibonacci and Similar Tree-Strips.” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4, Springer, 2014, pp. 409–40, doi:10.1007/s11040-014-9163-4. short: C. Sadel, Mathematical Physics, Analysis and Geometry 17 (2014) 409–440. date_created: 2018-12-11T11:54:45Z date_published: 2014-12-17T00:00:00Z date_updated: 2021-01-12T06:54:07Z day: '17' department: - _id: LaEr doi: 10.1007/s11040-014-9163-4 ec_funded: 1 external_id: arxiv: - '1304.3862' intvolume: ' 17' issue: 3-4 language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1304.3862 month: '12' oa: 1 oa_version: Preprint page: 409 - 440 project: - _id: 26450934-B435-11E9-9278-68D0E5697425 name: NSERC Postdoctoral fellowship - _id: 25681D80-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '291734' name: International IST Postdoc Fellowship Programme publication: Mathematical Physics, Analysis and Geometry publication_status: published publisher: Springer publist_id: '5168' quality_controlled: '1' scopus_import: 1 status: public title: Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 17 year: '2014' ... --- _id: '1937' abstract: - lang: eng text: We prove the edge universality of the beta ensembles for any β ≥ 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C4 and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class C4. author: - first_name: Paul full_name: Bourgade, Paul last_name: Bourgade - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Horngtzer full_name: Yau, Horngtzer last_name: Yau citation: ama: Bourgade P, Erdös L, Yau H. Edge universality of beta ensembles. Communications in Mathematical Physics. 2014;332(1):261-353. doi:10.1007/s00220-014-2120-z apa: Bourgade, P., Erdös, L., & Yau, H. (2014). Edge universality of beta ensembles. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2120-z chicago: Bourgade, Paul, László Erdös, and Horngtzer Yau. “Edge Universality of Beta Ensembles.” Communications in Mathematical Physics. Springer, 2014. https://doi.org/10.1007/s00220-014-2120-z. ieee: P. Bourgade, L. Erdös, and H. Yau, “Edge universality of beta ensembles,” Communications in Mathematical Physics, vol. 332, no. 1. Springer, pp. 261–353, 2014. ista: Bourgade P, Erdös L, Yau H. 2014. Edge universality of beta ensembles. Communications in Mathematical Physics. 332(1), 261–353. mla: Bourgade, Paul, et al. “Edge Universality of Beta Ensembles.” Communications in Mathematical Physics, vol. 332, no. 1, Springer, 2014, pp. 261–353, doi:10.1007/s00220-014-2120-z. short: P. Bourgade, L. Erdös, H. Yau, Communications in Mathematical Physics 332 (2014) 261–353. date_created: 2018-12-11T11:54:48Z date_published: 2014-11-01T00:00:00Z date_updated: 2021-01-12T06:54:12Z day: '01' department: - _id: LaEr doi: 10.1007/s00220-014-2120-z intvolume: ' 332' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1306.5728 month: '11' oa: 1 oa_version: Submitted Version page: 261 - 353 project: - _id: 25BDE9A4-B435-11E9-9278-68D0E5697425 grant_number: SFB-TR3-TP10B name: Glutamaterge synaptische Übertragung und Plastizität in hippocampalen Mikroschaltkreisen publication: Communications in Mathematical Physics publication_status: published publisher: Springer publist_id: '5158' quality_controlled: '1' scopus_import: 1 status: public title: Edge universality of beta ensembles type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 332 year: '2014' ... --- _id: '2019' abstract: - lang: eng text: We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of Keating et al. (2014) that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for p-uniform hypergraphs that correspond to p-spin glass Hamiltonians acting on n distinguishable spin- 1/2 particles. At the critical threshold p = n1/2 we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory. author: - 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 last_name: Schröder citation: ama: Erdös L, Schröder DJ. Phase transition in the density of states of quantum spin glasses. Mathematical Physics, Analysis and Geometry. 2014;17(3-4):441-464. doi:10.1007/s11040-014-9164-3 apa: Erdös, L., & Schröder, D. J. (2014). Phase transition in the density of states of quantum spin glasses. Mathematical Physics, Analysis and Geometry. Springer. https://doi.org/10.1007/s11040-014-9164-3 chicago: Erdös, László, and Dominik J Schröder. “Phase Transition in the Density of States of Quantum Spin Glasses.” Mathematical Physics, Analysis and Geometry. Springer, 2014. https://doi.org/10.1007/s11040-014-9164-3. ieee: L. Erdös and D. J. Schröder, “Phase transition in the density of states of quantum spin glasses,” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4. Springer, pp. 441–464, 2014. ista: Erdös L, Schröder DJ. 2014. Phase transition in the density of states of quantum spin glasses. Mathematical Physics, Analysis and Geometry. 17(3–4), 441–464. mla: Erdös, László, and Dominik J. Schröder. “Phase Transition in the Density of States of Quantum Spin Glasses.” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4, Springer, 2014, pp. 441–64, doi:10.1007/s11040-014-9164-3. short: L. Erdös, D.J. Schröder, Mathematical Physics, Analysis and Geometry 17 (2014) 441–464. date_created: 2018-12-11T11:55:15Z date_published: 2014-12-17T00:00:00Z date_updated: 2021-01-12T06:54:45Z day: '17' department: - _id: LaEr doi: 10.1007/s11040-014-9164-3 ec_funded: 1 intvolume: ' 17' issue: 3-4 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1407.1552 month: '12' oa: 1 oa_version: Submitted Version page: 441 - 464 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Mathematical Physics, Analysis and Geometry publication_status: published publisher: Springer publist_id: '5053' quality_controlled: '1' scopus_import: 1 status: public title: Phase transition in the density of states of quantum spin glasses type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 17 year: '2014' ... --- _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' ... --- _id: '2225' abstract: - lang: eng text: "We consider sample covariance matrices of the form X∗X, where X is an M×N matrix with independent random entries. We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent (X∗X−z)−1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.\r\n" article_number: '33' author: - first_name: Alex full_name: Bloemendal, Alex last_name: Bloemendal - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Antti full_name: Knowles, Antti last_name: Knowles - first_name: Horng full_name: Yau, Horng last_name: Yau - first_name: Jun full_name: Yin, Jun last_name: Yin citation: ama: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 2014;19. doi:10.1214/EJP.v19-3054 apa: Bloemendal, A., Erdös, L., Knowles, A., Yau, H., & Yin, J. (2014). Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v19-3054 chicago: Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2014. https://doi.org/10.1214/EJP.v19-3054. ieee: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local laws for sample covariance and generalized Wigner matrices,” Electronic Journal of Probability, vol. 19. Institute of Mathematical Statistics, 2014. ista: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. 2014. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 19, 33. mla: Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability, vol. 19, 33, Institute of Mathematical Statistics, 2014, doi:10.1214/EJP.v19-3054. short: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014). date_created: 2018-12-11T11:56:25Z date_published: 2014-03-15T00:00:00Z date_updated: 2021-01-12T06:56:07Z day: '15' ddc: - '510' department: - _id: LaEr doi: 10.1214/EJP.v19-3054 ec_funded: 1 file: - access_level: open_access checksum: 7eb297ff367a2ee73b21b6dd1e1948e4 content_type: application/pdf creator: system date_created: 2018-12-12T10:14:06Z date_updated: 2020-07-14T12:45:34Z file_id: '5055' file_name: IST-2016-427-v1+1_3054-16624-4-PB.pdf file_size: 810150 relation: main_file file_date_updated: 2020-07-14T12:45:34Z has_accepted_license: '1' intvolume: ' 19' language: - iso: eng month: '03' 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: Electronic Journal of Probability publication_identifier: issn: - '10836489' publication_status: published publisher: Institute of Mathematical Statistics publist_id: '4739' pubrep_id: '427' quality_controlled: '1' status: public title: Isotropic local laws for sample covariance and generalized 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 19 year: '2014' ... --- _id: '2699' abstract: - lang: eng text: "We prove the universality of the β-ensembles with convex analytic potentials and for any β >\r\n0, i.e. we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles." author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Paul full_name: Bourgade, Paul last_name: Bourgade - first_name: Horng full_name: Yau, Horng last_name: Yau citation: ama: Erdös L, Bourgade P, Yau H. Universality of general β-ensembles. Duke Mathematical Journal. 2014;163(6):1127-1190. doi:10.1215/00127094-2649752 apa: Erdös, L., Bourgade, P., & Yau, H. (2014). Universality of general β-ensembles. Duke Mathematical Journal. Duke University Press. https://doi.org/10.1215/00127094-2649752 chicago: Erdös, László, Paul Bourgade, and Horng Yau. “Universality of General β-Ensembles.” Duke Mathematical Journal. Duke University Press, 2014. https://doi.org/10.1215/00127094-2649752. ieee: L. Erdös, P. Bourgade, and H. Yau, “Universality of general β-ensembles,” Duke Mathematical Journal, vol. 163, no. 6. Duke University Press, pp. 1127–1190, 2014. ista: Erdös L, Bourgade P, Yau H. 2014. Universality of general β-ensembles. Duke Mathematical Journal. 163(6), 1127–1190. mla: Erdös, László, et al. “Universality of General β-Ensembles.” Duke Mathematical Journal, vol. 163, no. 6, Duke University Press, 2014, pp. 1127–90, doi:10.1215/00127094-2649752. short: L. Erdös, P. Bourgade, H. Yau, Duke Mathematical Journal 163 (2014) 1127–1190. date_created: 2018-12-11T11:59:08Z date_published: 2014-04-01T00:00:00Z date_updated: 2021-01-12T06:59:07Z day: '01' department: - _id: LaEr doi: 10.1215/00127094-2649752 intvolume: ' 163' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1104.2272 month: '04' oa: 1 oa_version: Preprint page: 1127 - 1190 publication: Duke Mathematical Journal publication_status: published publisher: Duke University Press publist_id: '4197' quality_controlled: '1' scopus_import: 1 status: public title: Universality of general β-ensembles type: journal_article user_id: 3FFCCD3A-F248-11E8-B48F-1D18A9856A87 volume: 163 year: '2014' ... --- _id: '1507' abstract: - lang: eng text: The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices. acknowledgement: The author is partially supported by SFB-TR 12 Grant of the German Research Council. article_processing_charge: No author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 citation: ama: 'Erdös L. Random matrices, log-gases and Hölder regularity. In: Proceedings of the International Congress of Mathematicians. Vol 3. International Congress of Mathematicians; 2014:214-236.' apa: 'Erdös, L. (2014). Random matrices, log-gases and Hölder regularity. In Proceedings of the International Congress of Mathematicians (Vol. 3, pp. 214–236). Seoul, Korea: International Congress of Mathematicians.' chicago: Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” In Proceedings of the International Congress of Mathematicians, 3:214–36. International Congress of Mathematicians, 2014. ieee: L. Erdös, “Random matrices, log-gases and Hölder regularity,” in Proceedings of the International Congress of Mathematicians, Seoul, Korea, 2014, vol. 3, pp. 214–236. ista: 'Erdös L. 2014. Random matrices, log-gases and Hölder regularity. Proceedings of the International Congress of Mathematicians. ICM: International Congress of Mathematicians vol. 3, 214–236.' mla: Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” Proceedings of the International Congress of Mathematicians, vol. 3, International Congress of Mathematicians, 2014, pp. 214–36. short: L. Erdös, in:, Proceedings of the International Congress of Mathematicians, International Congress of Mathematicians, 2014, pp. 214–236. conference: end_date: 2014-08-21 location: Seoul, Korea name: 'ICM: International Congress of Mathematicians' start_date: 2014-08-13 date_created: 2018-12-11T11:52:25Z date_published: 2014-08-01T00:00:00Z date_updated: 2023-10-17T11:12:55Z day: '01' department: - _id: LaEr ec_funded: 1 intvolume: ' 3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1407.5752 month: '08' oa: 1 oa_version: Submitted Version page: 214 - 236 project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Proceedings of the International Congress of Mathematicians publication_status: published publisher: International Congress of Mathematicians publist_id: '5670' quality_controlled: '1' scopus_import: '1' status: public title: Random matrices, log-gases and Hölder regularity type: conference user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 3 year: '2014' ... --- _id: '2698' abstract: - lang: eng text: We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B. The total energy includes the field energy β∫B2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, h→0, of the total ground state energy E(β,h,V). The relevant parameter measuring the field strength in the semiclassical limit is κ=βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in κ or even for fixed κ. We do however give upper and lower bounds on E with almost matching dependence on κ. In the simultaneous limit h→0 and κ→∞ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schrödinger operator. author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Søren full_name: Fournais, Søren last_name: Fournais - first_name: Jan full_name: Solovej, Jan last_name: Solovej citation: ama: Erdös L, Fournais S, Solovej J. Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. 2013;15(6):2093-2113. doi:10.4171/JEMS/416 apa: Erdös, L., Fournais, S., & Solovej, J. (2013). Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. European Mathematical Society. https://doi.org/10.4171/JEMS/416 chicago: Erdös, László, Søren Fournais, and Jan Solovej. “Stability and Semiclassics in Self-Generated Fields.” Journal of the European Mathematical Society. European Mathematical Society, 2013. https://doi.org/10.4171/JEMS/416. ieee: L. Erdös, S. Fournais, and J. Solovej, “Stability and semiclassics in self-generated fields,” Journal of the European Mathematical Society, vol. 15, no. 6. European Mathematical Society, pp. 2093–2113, 2013. ista: Erdös L, Fournais S, Solovej J. 2013. Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. 15(6), 2093–2113. mla: Erdös, László, et al. “Stability and Semiclassics in Self-Generated Fields.” Journal of the European Mathematical Society, vol. 15, no. 6, European Mathematical Society, 2013, pp. 2093–113, doi:10.4171/JEMS/416. short: L. Erdös, S. Fournais, J. Solovej, Journal of the European Mathematical Society 15 (2013) 2093–2113. date_created: 2018-12-11T11:59:07Z date_published: 2013-10-16T00:00:00Z date_updated: 2021-01-12T06:59:07Z day: '16' department: - _id: LaEr doi: 10.4171/JEMS/416 external_id: arxiv: - '1105.0506' intvolume: ' 15' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1105.0506 month: '10' oa: 1 oa_version: Preprint page: 2093 - 2113 publication: Journal of the European Mathematical Society publication_status: published publisher: European Mathematical Society publist_id: '4198' quality_controlled: '1' status: public title: Stability and semiclassics in self-generated fields type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 15 year: '2013' ... --- _id: '2782' abstract: - lang: eng text: We consider random n×n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices. author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Brendan full_name: Farrell, Brendan last_name: Farrell citation: ama: Erdös L, Farrell B. Local eigenvalue density for general MANOVA matrices. Journal of Statistical Physics. 2013;152(6):1003-1032. doi:10.1007/s10955-013-0807-8 apa: Erdös, L., & Farrell, B. (2013). Local eigenvalue density for general MANOVA matrices. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-013-0807-8 chicago: Erdös, László, and Brendan Farrell. “Local Eigenvalue Density for General MANOVA Matrices.” Journal of Statistical Physics. Springer, 2013. https://doi.org/10.1007/s10955-013-0807-8. ieee: L. Erdös and B. Farrell, “Local eigenvalue density for general MANOVA matrices,” Journal of Statistical Physics, vol. 152, no. 6. Springer, pp. 1003–1032, 2013. ista: Erdös L, Farrell B. 2013. Local eigenvalue density for general MANOVA matrices. Journal of Statistical Physics. 152(6), 1003–1032. mla: Erdös, László, and Brendan Farrell. “Local Eigenvalue Density for General MANOVA Matrices.” Journal of Statistical Physics, vol. 152, no. 6, Springer, 2013, pp. 1003–32, doi:10.1007/s10955-013-0807-8. short: L. Erdös, B. Farrell, Journal of Statistical Physics 152 (2013) 1003–1032. date_created: 2018-12-11T11:59:34Z date_published: 2013-07-18T00:00:00Z date_updated: 2021-01-12T06:59:41Z day: '18' department: - _id: LaEr doi: 10.1007/s10955-013-0807-8 external_id: arxiv: - '1207.0031' intvolume: ' 152' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1207.0031 month: '07' oa: 1 oa_version: Preprint page: 1003 - 1032 publication: Journal of Statistical Physics publication_status: published publisher: Springer publist_id: '4107' quality_controlled: '1' scopus_import: 1 status: public title: Local eigenvalue density for general MANOVA matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 152 year: '2013' ... --- _id: '2837' abstract: - lang: eng text: We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij|2. As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W ≫N1-εn with some εn > 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17, 19, 6]. author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Antti full_name: Knowles, Antti last_name: Knowles - first_name: Horng full_name: Yau, Horng last_name: Yau - first_name: Jun full_name: Yin, Jun last_name: Yin citation: ama: Erdös L, Knowles A, Yau H, Yin J. The local semicircle law for a general class of random matrices. Electronic Journal of Probability. 2013;18(59):1-58. doi:10.1214/EJP.v18-2473 apa: Erdös, L., Knowles, A., Yau, H., & Yin, J. (2013). The local semicircle law for a general class of random matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v18-2473 chicago: Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “The Local Semicircle Law for a General Class of Random Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2013. https://doi.org/10.1214/EJP.v18-2473. ieee: L. Erdös, A. Knowles, H. Yau, and J. Yin, “The local semicircle law for a general class of random matrices,” Electronic Journal of Probability, vol. 18, no. 59. Institute of Mathematical Statistics, pp. 1–58, 2013. ista: Erdös L, Knowles A, Yau H, Yin J. 2013. The local semicircle law for a general class of random matrices. Electronic Journal of Probability. 18(59), 1–58. mla: Erdös, László, et al. “The Local Semicircle Law for a General Class of Random Matrices.” Electronic Journal of Probability, vol. 18, no. 59, Institute of Mathematical Statistics, 2013, pp. 1–58, doi:10.1214/EJP.v18-2473. short: L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 18 (2013) 1–58. date_created: 2018-12-11T11:59:51Z date_published: 2013-05-29T00:00:00Z date_updated: 2021-01-12T07:00:06Z day: '29' ddc: - '530' department: - _id: LaEr doi: 10.1214/EJP.v18-2473 file: - access_level: open_access checksum: aac9e52a00cb2f5149dc9e362b5ccf44 content_type: application/pdf creator: system date_created: 2018-12-12T10:15:46Z date_updated: 2020-07-14T12:45:50Z file_id: '5169' file_name: IST-2016-406-v1+1_2473-13759-1-PB.pdf file_size: 651497 relation: main_file file_date_updated: 2020-07-14T12:45:50Z has_accepted_license: '1' intvolume: ' 18' issue: '59' language: - iso: eng month: '05' oa: 1 oa_version: Published Version page: 1-58 publication: Electronic Journal of Probability publication_status: published publisher: Institute of Mathematical Statistics publist_id: '3962' pubrep_id: '406' quality_controlled: '1' scopus_import: 1 status: public title: The local semicircle law for a general class of random 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 18 year: '2013' ...