[{"type":"journal_article","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices."}],"month":"08","_id":"483","day":"25","ec_funded":1,"department":[{"_id":"LaEr"}],"publist_id":"7337","publication":"Advances in Theoretical and Mathematical Physics","volume":21,"oa":1,"title":"Universality for a class of random band matrices","scopus_import":1,"oa_version":"Submitted Version","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2017-08-25T00:00:00Z","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"338804"}],"doi":"10.4310/ATMP.2017.v21.n3.a5","publication_status":"published","year":"2017","author":[{"first_name":"Paul","last_name":"Bourgade","full_name":"Bourgade, Paul"},{"full_name":"Erdös, László","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"quality_controlled":"1","page":"739 - 800","date_updated":"2021-01-12T08:00:57Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1602.02312"}],"citation":{"mla":"Bourgade, Paul, et al. “Universality for a Class of Random Band Matrices.” *Advances in Theoretical and Mathematical Physics*, vol. 21, no. 3, International Press, 2017, pp. 739–800, doi:10.4310/ATMP.2017.v21.n3.a5.","chicago":"Bourgade, Paul, László Erdös, Horng Yau, and Jun Yin. “Universality for a Class of Random Band Matrices.” *Advances in Theoretical and Mathematical Physics*. International Press, 2017. https://doi.org/10.4310/ATMP.2017.v21.n3.a5.","ista":"Bourgade P, Erdös L, Yau H, Yin J. 2017. Universality for a class of random band matrices. Advances in Theoretical and Mathematical Physics. 21(3), 739–800.","short":"P. Bourgade, L. Erdös, H. Yau, J. Yin, Advances in Theoretical and Mathematical Physics 21 (2017) 739–800.","apa":"Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2017). Universality for a class of random band matrices. *Advances in Theoretical and Mathematical Physics*. International Press. https://doi.org/10.4310/ATMP.2017.v21.n3.a5","ieee":"P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Universality for a class of random band matrices,” *Advances in Theoretical and Mathematical Physics*, vol. 21, no. 3. International Press, pp. 739–800, 2017.","ama":"Bourgade P, Erdös L, Yau H, Yin J. Universality for a class of random band matrices. *Advances in Theoretical and Mathematical Physics*. 2017;21(3):739-800. doi:10.4310/ATMP.2017.v21.n3.a5"},"publication_identifier":{"issn":["10950761"]},"date_created":"2018-12-11T11:46:43Z","publisher":"International Press","issue":"3","intvolume":" 21","status":"public"},{"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","ieee":"J. Alt, “Singularities of the density of states of random Gram matrices,” *Electronic Communications in Probability*, vol. 22. Institute of Mathematical Statistics, 2017.","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","short":"J. Alt, Electronic Communications in Probability 22 (2017).","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.","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."},"publication_identifier":{"issn":["1083589X"]},"ddc":["539"],"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"date_created":"2018-12-11T11:47:07Z","date_updated":"2021-01-12T08:02:34Z","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"149"}]},"publisher":"Institute of Mathematical Statistics","intvolume":" 22","article_number":"63","status":"public","publist_id":"7265","publication":"Electronic Communications in Probability","oa":1,"volume":22,"has_accepted_license":"1","title":"Singularities of the density of states of random Gram matrices","scopus_import":1,"type":"journal_article","language":[{"iso":"eng"}],"file":[{"date_updated":"2020-07-14T12:47:00Z","file_id":"4663","creator":"system","content_type":"application/pdf","relation":"main_file","file_name":"IST-2018-926-v1+1_euclid.ecp.1511233247.pdf","access_level":"open_access","checksum":"0ec05303a0de190de145654237984c79","file_size":470876,"date_created":"2018-12-12T10:08:04Z"}],"abstract":[{"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.","lang":"eng"}],"_id":"550","month":"11","day":"21","ec_funded":1,"department":[{"_id":"LaEr"}],"publication_status":"published","year":"2017","author":[{"last_name":"Alt","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes"}],"quality_controlled":"1","oa_version":"Published Version","date_published":"2017-11-21T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"call_identifier":"FP7","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"file_date_updated":"2020-07-14T12:47:00Z","doi":"10.1214/17-ECP97","pubrep_id":"926"},{"date_created":"2018-12-11T11:47:13Z","title":"A dynamical approach to random matrix theory","volume":28,"citation":{"ieee":"L. Erdös and H. Yau, *A dynamical approach to random matrix theory*, vol. 28. American Mathematical Society, 2017.","apa":"Erdös, L., & Yau, H. (2017). *A dynamical approach to random matrix theory* (Vol. 28). American Mathematical Society.","ama":"Erdös L, Yau H. *A Dynamical Approach to Random Matrix Theory*. Vol 28. American Mathematical Society; 2017.","mla":"Erdös, László, and Horng Yau. *A Dynamical Approach to Random Matrix Theory*. Vol. 28, American Mathematical Society, 2017.","ista":"Erdös L, Yau H. 2017. A dynamical approach to random matrix theory, American Mathematical Society, 226p.","chicago":"Erdös, László, and Horng Yau. *A Dynamical Approach to Random Matrix Theory*. Vol. 28. Courant Lecture Notes. American Mathematical Society, 2017.","short":"L. Erdös, H. Yau, A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 2017."},"publication_identifier":{"isbn":["9781470436483"],"eisbn":["978-1-4704-4194-4"]},"publist_id":"7247","ec_funded":1,"department":[{"_id":"LaEr"}],"day":"01","_id":"567","month":"01","series_title":"Courant Lecture Notes","date_updated":"2020-01-16T12:37:45Z","abstract":[{"lang":"eng","text":"This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.\r\n"}],"type":"book","page":"226","language":[{"iso":"eng"}],"alternative_title":["Courant Lecture Notes"],"quality_controlled":"1","year":"2017","author":[{"full_name":"Erdös, László","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603"},{"full_name":"Yau, Horng","last_name":"Yau","first_name":"Horng"}],"publication_status":"published","status":"public","date_published":"2017-01-01T00:00:00Z","project":[{"grant_number":"338804","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"intvolume":" 28","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"None","publisher":"American Mathematical Society"},{"issue":"4","publisher":"Institute of Mathematical Statistics","intvolume":" 53","status":"public","publication_identifier":{"issn":["02460203"]},"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","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.","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.","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.","short":"L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 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."},"date_created":"2018-12-11T11:47:30Z","page":"1606 - 1656","date_updated":"2021-01-12T08:06:22Z","main_file_link":[{"url":"https://arxiv.org/abs/1504.00650","open_access":"1"}],"publication_status":"published","author":[{"orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","full_name":"Erdös, László"},{"orcid":"0000-0003-0954-3231","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","full_name":"Schnelli, Kevin"}],"year":"2017","quality_controlled":"1","oa_version":"Submitted Version","date_published":"2017-11-01T00:00:00Z","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","doi":"10.1214/16-AIHP765","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","publist_id":"7189","oa":1,"volume":53,"scopus_import":1,"title":"Universality for random matrix flows with time dependent density","language":[{"iso":"eng"}],"type":"journal_article","abstract":[{"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.","lang":"eng"}],"_id":"615","month":"11","ec_funded":1,"department":[{"_id":"LaEr"}],"day":"01"},{"issue":"3-4","publisher":"Springer","intvolume":" 169","status":"public","publication_identifier":{"issn":["01788051"]},"citation":{"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.","ista":"Ajanki OH, Erdös L, Krüger TH. 2017. Universality for general Wigner-type matrices. Probability Theory and Related Fields. 169(3–4), 667–727.","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.","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","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"},"ddc":["510","530"],"date_created":"2018-12-11T11:51:27Z","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"page":"667 - 727","date_updated":"2021-01-12T06:49:58Z","publication_status":"published","author":[{"first_name":"Oskari H","last_name":"Ajanki","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","full_name":"Ajanki, Oskari H"},{"full_name":"Erdös, László","first_name":"László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603"},{"full_name":"Krüger, Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger","first_name":"Torben H","orcid":"0000-0002-4821-3297"}],"year":"2017","quality_controlled":"1","oa_version":"Published Version","article_processing_charge":"Yes (via OA deal)","file_date_updated":"2020-07-14T12:44:44Z","date_published":"2017-12-01T00:00:00Z","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804"},{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","pubrep_id":"657","doi":"10.1007/s00440-016-0740-2","publist_id":"5930","publication":"Probability Theory and Related Fields","oa":1,"volume":169,"title":"Universality for general Wigner-type matrices","scopus_import":1,"has_accepted_license":"1","language":[{"iso":"eng"}],"type":"journal_article","abstract":[{"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.","lang":"eng"}],"file":[{"creator":"system","access_level":"open_access","file_name":"IST-2017-657-v1+2_s00440-016-0740-2.pdf","relation":"main_file","content_type":"application/pdf","date_updated":"2020-07-14T12:44:44Z","file_id":"4686","date_created":"2018-12-12T10:08:25Z","file_size":988843,"checksum":"29f5a72c3f91e408aeb9e78344973803"}],"month":"12","_id":"1337","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). ","day":"01","ec_funded":1,"department":[{"_id":"LaEr"}]}]