[{"department":[{"_id":"GradSch"},{"_id":"LaEr"}],"file_date_updated":"2021-01-25T14:19:10Z","ddc":["510"],"supervisor":[{"last_name":"Erdös","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"}],"date_updated":"2023-09-07T13:29:32Z","status":"public","type":"dissertation","_id":"9022","ec_funded":1,"file":[{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"file_id":"9043","checksum":"5a93658a5f19478372523ee232887e2b","file_size":4127796,"date_updated":"2021-01-25T14:19:03Z","creator":"gcipollo","file_name":"thesis.pdf","date_created":"2021-01-25T14:19:03Z"},{"access_level":"closed","relation":"source_file","content_type":"application/zip","checksum":"e8270eddfe6a988e92a53c88d1d19b8c","file_id":"9044","creator":"gcipollo","date_updated":"2021-01-25T14:19:10Z","file_size":12775206,"date_created":"2021-01-25T14:19:10Z","file_name":"Thesis_files.zip"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2663-337X"]},"publication_status":"published","degree_awarded":"PhD","month":"01","alternative_title":["ISTA Thesis"],"oa_version":"Published Version","abstract":[{"lang":"eng","text":"In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.\r\nIn the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time\r\n(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.\r\nIn the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result\r\nimproves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices."}],"title":"Fluctuations in the spectrum of random matrices","author":[{"first_name":"Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","last_name":"Cipolloni","orcid":"0000-0002-4901-7992","full_name":"Cipolloni, Giorgio"}],"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"short":"G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, Institute of Science and Technology Austria, 2021.","ieee":"G. Cipolloni, “Fluctuations in the spectrum of random matrices,” Institute of Science and Technology Austria, 2021.","apa":"Cipolloni, G. (2021). Fluctuations in the spectrum of random matrices. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:9022","ama":"Cipolloni G. Fluctuations in the spectrum of random matrices. 2021. doi:10.15479/AT:ISTA:9022","mla":"Cipolloni, Giorgio. Fluctuations in the Spectrum of Random Matrices. Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9022.","ista":"Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. Institute of Science and Technology Austria.","chicago":"Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.” Institute of Science and Technology Austria, 2021. https://doi.org/10.15479/AT:ISTA:9022."},"project":[{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385"},{"call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"date_published":"2021-01-25T00:00:00Z","doi":"10.15479/AT:ISTA:9022","date_created":"2021-01-21T18:16:54Z","page":"380","day":"25","has_accepted_license":"1","year":"2021","publisher":"Institute of Science and Technology Austria","oa":1,"acknowledgement":"I gratefully acknowledge the financial support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385 and my advisor’s ERC Advanced Grant No. 338804."},{"title":"From Dyson to Pearcey: Universal statistics in random matrix theory","author":[{"last_name":"Schröder","full_name":"Schröder, Dominik J","orcid":"0000-0002-2904-1856","first_name":"Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"article_processing_charge":"No","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"mla":"Schröder, Dominik J. From Dyson to Pearcey: Universal Statistics in Random Matrix Theory. Institute of Science and Technology Austria, 2019, doi:10.15479/AT:ISTA:th6179.","ama":"Schröder DJ. From Dyson to Pearcey: Universal statistics in random matrix theory. 2019. doi:10.15479/AT:ISTA:th6179","apa":"Schröder, D. J. (2019). From Dyson to Pearcey: Universal statistics in random matrix theory. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th6179","ieee":"D. J. Schröder, “From Dyson to Pearcey: Universal statistics in random matrix theory,” Institute of Science and Technology Austria, 2019.","short":"D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix Theory, Institute of Science and Technology Austria, 2019.","chicago":"Schröder, Dominik J. “From Dyson to Pearcey: Universal Statistics in Random Matrix Theory.” Institute of Science and Technology Austria, 2019. https://doi.org/10.15479/AT:ISTA:th6179.","ista":"Schröder DJ. 2019. From Dyson to Pearcey: Universal statistics in random matrix theory. Institute of Science and Technology Austria."},"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"doi":"10.15479/AT:ISTA:th6179","date_published":"2019-03-18T00:00:00Z","date_created":"2019-03-28T08:58:59Z","page":"375","day":"18","has_accepted_license":"1","year":"2019","publisher":"Institute of Science and Technology Austria","oa":1,"file_date_updated":"2020-07-14T12:47:21Z","department":[{"_id":"LaEr"}],"ddc":["515","519"],"supervisor":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"}],"date_updated":"2024-02-22T14:34:33Z","status":"public","type":"dissertation","_id":"6179","related_material":{"record":[{"relation":"part_of_dissertation","status":"public","id":"1144"},{"id":"6186","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"6185","status":"public"},{"id":"6182","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"1012","status":"public"},{"id":"6184","status":"public","relation":"part_of_dissertation"}]},"ec_funded":1,"file":[{"access_level":"closed","relation":"source_file","content_type":"application/x-gzip","checksum":"6926f66f28079a81c4937e3764be00fc","file_id":"6180","creator":"dernst","date_updated":"2020-07-14T12:47:21Z","file_size":7104482,"date_created":"2019-03-28T08:53:52Z","file_name":"2019_Schroeder_Thesis.tar.gz"},{"creator":"dernst","file_size":4228794,"date_updated":"2020-07-14T12:47:21Z","file_name":"2019_Schroeder_Thesis.pdf","date_created":"2019-03-28T08:53:52Z","relation":"main_file","access_level":"open_access","content_type":"application/pdf","file_id":"6181","checksum":"7d0ebb8d1207e89768cdd497a5bf80fb"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2663-337X"]},"publication_status":"published","degree_awarded":"PhD","month":"03","alternative_title":["ISTA Thesis"],"oa_version":"Published Version","abstract":[{"lang":"eng","text":"In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.\r\nIn the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.\r\nIn the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure."}]},{"project":[{"name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"ieee":"J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” Institute of Science and Technology Austria, 2018.","short":"J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, Institute of Science and Technology Austria, 2018.","apa":"Alt, J. (2018). Dyson equation and eigenvalue statistics of random matrices. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:TH_1040","ama":"Alt J. Dyson equation and eigenvalue statistics of random matrices. 2018. doi:10.15479/AT:ISTA:TH_1040","mla":"Alt, Johannes. Dyson Equation and Eigenvalue Statistics of Random Matrices. Institute of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:TH_1040.","ista":"Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices. Institute of Science and Technology Austria.","chicago":"Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.” Institute of Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:TH_1040."},"title":"Dyson equation and eigenvalue statistics of random matrices","publist_id":"7772","author":[{"id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes","full_name":"Alt, Johannes","last_name":"Alt"}],"article_processing_charge":"No","publisher":"Institute of Science and Technology Austria","oa":1,"day":"12","has_accepted_license":"1","year":"2018","doi":"10.15479/AT:ISTA:TH_1040","date_published":"2018-07-12T00:00:00Z","date_created":"2018-12-11T11:44:53Z","page":"456","_id":"149","status":"public","pubrep_id":"1040","type":"dissertation","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["515","519"],"supervisor":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","full_name":"Erdös, László","orcid":"0000-0001-5366-9603"}],"date_updated":"2024-02-22T14:34:33Z","file_date_updated":"2020-07-14T12:44:57Z","department":[{"_id":"LaEr"}],"oa_version":"Published Version","abstract":[{"text":"The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.","lang":"eng"}],"month":"07","alternative_title":["ISTA Thesis"],"file":[{"file_id":"6241","checksum":"d4dad55a7513f345706aaaba90cb1bb8","access_level":"open_access","relation":"main_file","content_type":"application/pdf","date_created":"2019-04-08T13:55:20Z","file_name":"2018_thesis_Alt.pdf","creator":"dernst","date_updated":"2020-07-14T12:44:57Z","file_size":5801709},{"content_type":"application/zip","relation":"source_file","access_level":"closed","checksum":"d73fcf46300dce74c403f2b491148ab4","file_id":"6242","file_size":3802059,"date_updated":"2020-07-14T12:44:57Z","creator":"dernst","file_name":"2018_thesis_Alt_source.zip","date_created":"2019-04-08T13:55:20Z"}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2663-337X"]},"degree_awarded":"PhD","publication_status":"published","related_material":{"record":[{"status":"public","id":"1677","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"550"},{"status":"public","id":"6183","relation":"part_of_dissertation"},{"status":"public","id":"566","relation":"part_of_dissertation"},{"status":"public","id":"1010","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"6240"},{"status":"public","id":"6184","relation":"part_of_dissertation"}]},"license":"https://creativecommons.org/licenses/by/4.0/","ec_funded":1}]