[{"alternative_title":["ISTA Thesis"],"type":"dissertation","abstract":[{"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.","lang":"eng"}],"ddc":["510"],"status":"public","title":"Fluctuations in the spectrum of random matrices","_id":"9022","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"Published Version","file":[{"file_name":"thesis.pdf","access_level":"open_access","content_type":"application/pdf","file_size":4127796,"creator":"gcipollo","relation":"main_file","file_id":"9043","date_created":"2021-01-25T14:19:03Z","date_updated":"2021-01-25T14:19:03Z","checksum":"5a93658a5f19478372523ee232887e2b","success":1},{"file_name":"Thesis_files.zip","access_level":"closed","creator":"gcipollo","content_type":"application/zip","file_size":12775206,"file_id":"9044","relation":"source_file","date_created":"2021-01-25T14:19:10Z","date_updated":"2021-01-25T14:19:10Z","checksum":"e8270eddfe6a988e92a53c88d1d19b8c"}],"day":"25","has_accepted_license":"1","article_processing_charge":"No","page":"380","citation":{"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","ista":"Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. Institute of Science and Technology Austria.","ama":"Cipolloni G. Fluctuations in the spectrum of random matrices. 2021. doi:10.15479/AT:ISTA:9022","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.","short":"G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, Institute of Science and Technology Austria, 2021.","mla":"Cipolloni, Giorgio. Fluctuations in the Spectrum of Random Matrices. Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9022."},"date_published":"2021-01-25T00:00:00Z","file_date_updated":"2021-01-25T14:19:10Z","ec_funded":1,"publication_status":"published","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"publisher":"Institute of Science and Technology Austria","year":"2021","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.","date_updated":"2023-09-07T13:29:32Z","date_created":"2021-01-21T18:16:54Z","author":[{"id":"42198EFA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4901-7992","first_name":"Giorgio","last_name":"Cipolloni","full_name":"Cipolloni, Giorgio"}],"month":"01","publication_identifier":{"issn":["2663-337X"]},"project":[{"call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385","_id":"2564DBCA-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"oa":1,"supervisor":[{"full_name":"Erdös, László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","first_name":"László"}],"degree_awarded":"PhD","language":[{"iso":"eng"}],"doi":"10.15479/AT:ISTA:9022"},{"month":"05","publication_identifier":{"eissn":["2690-1005"],"issn":["2690-0998"]},"doi":"10.2140/pmp.2021.2.221","language":[{"iso":"eng"}],"external_id":{"arxiv":["1907.13631"]},"oa":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1907.13631","open_access":"1"}],"quality_controlled":"1","project":[{"call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"}],"ec_funded":1,"author":[{"last_name":"Alt","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes"},{"full_name":"Erdös, László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","first_name":"László"},{"first_name":"Torben H","last_name":"Krüger","id":"3020C786-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H"}],"date_updated":"2024-02-19T08:30:00Z","date_created":"2024-02-18T23:01:03Z","volume":2,"acknowledgement":"Partially supported by ERC Starting Grant RandMat No. 715539 and the SwissMap grant of Swiss National Science Foundation. Partially supported by ERC Advanced Grant RanMat No. 338804. Partially supported by the Hausdorff Center for Mathematics in Bonn.","year":"2021","publication_status":"published","publisher":"Mathematical Sciences Publishers","department":[{"_id":"LaEr"}],"day":"21","article_processing_charge":"No","scopus_import":"1","date_published":"2021-05-21T00:00:00Z","publication":"Probability and Mathematical Physics","citation":{"ista":"Alt J, Erdös L, Krüger TH. 2021. Spectral radius of random matrices with independent entries. Probability and Mathematical Physics. 2(2), 221–280.","apa":"Alt, J., Erdös, L., & Krüger, T. H. (2021). Spectral radius of random matrices with independent entries. Probability and Mathematical Physics. Mathematical Sciences Publishers. https://doi.org/10.2140/pmp.2021.2.221","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “Spectral radius of random matrices with independent entries,” Probability and Mathematical Physics, vol. 2, no. 2. Mathematical Sciences Publishers, pp. 221–280, 2021.","ama":"Alt J, Erdös L, Krüger TH. Spectral radius of random matrices with independent entries. Probability and Mathematical Physics. 2021;2(2):221-280. doi:10.2140/pmp.2021.2.221","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “Spectral Radius of Random Matrices with Independent Entries.” Probability and Mathematical Physics. Mathematical Sciences Publishers, 2021. https://doi.org/10.2140/pmp.2021.2.221.","mla":"Alt, Johannes, et al. “Spectral Radius of Random Matrices with Independent Entries.” Probability and Mathematical Physics, vol. 2, no. 2, Mathematical Sciences Publishers, 2021, pp. 221–80, doi:10.2140/pmp.2021.2.221.","short":"J. Alt, L. Erdös, T.H. Krüger, Probability and Mathematical Physics 2 (2021) 221–280."},"article_type":"original","page":"221-280","abstract":[{"lang":"eng","text":"We consider random n×n matrices X with independent and centered entries and a general variance profile. We show that the spectral radius of X converges with very high probability to the square root of the spectral radius of the variance matrix of X when n tends to infinity. We also establish the optimal rate of convergence, that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law [arXiv:1612.07776] at the spectral edge."}],"issue":"2","type":"journal_article","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"15013","title":"Spectral radius of random matrices with independent entries","status":"public","intvolume":" 2"},{"file":[{"file_name":"2020_ProbTheory_Cipolloni.pdf","access_level":"open_access","file_size":497032,"content_type":"application/pdf","creator":"dernst","relation":"main_file","file_id":"8612","date_created":"2020-10-05T14:53:40Z","date_updated":"2020-10-05T14:53:40Z","checksum":"611ae28d6055e1e298d53a57beb05ef4","success":1}],"oa_version":"Published Version","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","_id":"8601","status":"public","title":"Edge universality for non-Hermitian random matrices","ddc":["510"],"abstract":[{"lang":"eng","text":"We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble."}],"type":"journal_article","date_published":"2021-02-01T00:00:00Z","publication":"Probability Theory and Related Fields","citation":{"mla":"Cipolloni, Giorgio, et al. “Edge Universality for Non-Hermitian Random Matrices.” Probability Theory and Related Fields, Springer Nature, 2021, doi:10.1007/s00440-020-01003-7.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Probability Theory and Related Fields (2021).","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Edge Universality for Non-Hermitian Random Matrices.” Probability Theory and Related Fields. Springer Nature, 2021. https://doi.org/10.1007/s00440-020-01003-7.","ama":"Cipolloni G, Erdös L, Schröder DJ. Edge universality for non-Hermitian random matrices. Probability Theory and Related Fields. 2021. doi:10.1007/s00440-020-01003-7","ista":"Cipolloni G, Erdös L, Schröder DJ. 2021. Edge universality for non-Hermitian random matrices. Probability Theory and Related Fields.","apa":"Cipolloni, G., Erdös, L., & Schröder, D. J. (2021). Edge universality for non-Hermitian random matrices. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-020-01003-7","ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Edge universality for non-Hermitian random matrices,” Probability Theory and Related Fields. Springer Nature, 2021."},"article_type":"original","day":"01","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","scopus_import":"1","author":[{"full_name":"Cipolloni, Giorgio","last_name":"Cipolloni","first_name":"Giorgio","orcid":"0000-0002-4901-7992","id":"42198EFA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Erdös, László","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","first_name":"László"},{"full_name":"Schröder, Dominik J","last_name":"Schröder","first_name":"Dominik J","orcid":"0000-0002-2904-1856","id":"408ED176-F248-11E8-B48F-1D18A9856A87"}],"date_created":"2020-10-04T22:01:37Z","date_updated":"2024-03-07T15:07:53Z","year":"2021","publication_status":"published","department":[{"_id":"LaEr"}],"publisher":"Springer Nature","file_date_updated":"2020-10-05T14:53:40Z","ec_funded":1,"doi":"10.1007/s00440-020-01003-7","language":[{"iso":"eng"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"arxiv":["1908.00969"],"isi":["000572724600002"]},"oa":1,"isi":1,"quality_controlled":"1","project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"},{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"},{"call_identifier":"H2020","name":"International IST Doctoral Program","grant_number":"665385","_id":"2564DBCA-B435-11E9-9278-68D0E5697425"}],"month":"02","publication_identifier":{"eissn":["14322064"],"issn":["01788051"]}},{"month":"08","publication_identifier":{"issn":["00029947"],"eissn":["10886850"]},"isi":1,"quality_controlled":"1","project":[{"name":"Geometric study of Wasserstein spaces and free probability","call_identifier":"H2020","grant_number":"846294","_id":"26A455A6-B435-11E9-9278-68D0E5697425"}],"oa":1,"external_id":{"arxiv":["2002.00859"],"isi":["000551418100018"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2002.00859"}],"language":[{"iso":"eng"}],"doi":"10.1090/tran/8113","ec_funded":1,"publication_status":"published","department":[{"_id":"LaEr"}],"publisher":"American Mathematical Society","year":"2020","date_updated":"2023-08-17T14:31:03Z","date_created":"2020-01-29T10:20:46Z","volume":373,"author":[{"full_name":"Geher, Gyorgy Pal","first_name":"Gyorgy Pal","last_name":"Geher"},{"last_name":"Titkos","first_name":"Tamas","full_name":"Titkos, Tamas"},{"last_name":"Virosztek","first_name":"Daniel","orcid":"0000-0003-1109-5511","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","full_name":"Virosztek, Daniel"}],"keyword":["Wasserstein space","isometric embeddings","isometric rigidity","exotic isometry flow"],"day":"01","article_processing_charge":"No","article_type":"original","page":"5855-5883","publication":"Transactions of the American Mathematical Society","citation":{"ama":"Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society. 2020;373(8):5855-5883. doi:10.1090/tran/8113","ieee":"G. P. Geher, T. Titkos, and D. Virosztek, “Isometric study of Wasserstein spaces - the real line,” Transactions of the American Mathematical Society, vol. 373, no. 8. American Mathematical Society, pp. 5855–5883, 2020.","apa":"Geher, G. P., Titkos, T., & Virosztek, D. (2020). Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/tran/8113","ista":"Geher GP, Titkos T, Virosztek D. 2020. Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society. 373(8), 5855–5883.","short":"G.P. Geher, T. Titkos, D. Virosztek, Transactions of the American Mathematical Society 373 (2020) 5855–5883.","mla":"Geher, Gyorgy Pal, et al. “Isometric Study of Wasserstein Spaces - the Real Line.” Transactions of the American Mathematical Society, vol. 373, no. 8, American Mathematical Society, 2020, pp. 5855–83, doi:10.1090/tran/8113.","chicago":"Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Isometric Study of Wasserstein Spaces - the Real Line.” Transactions of the American Mathematical Society. American Mathematical Society, 2020. https://doi.org/10.1090/tran/8113."},"date_published":"2020-08-01T00:00:00Z","type":"journal_article","abstract":[{"text":"Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space\r\nW_p(R) for all p \\in [1,\\infty) \\setminus {2}. We show that W_2(R) is also exceptional regarding the\r\nparameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying\r\nspace, we prove that the exceptionality of p = 2 disappears if we replace R by the compact\r\ninterval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if\r\np is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))\r\ncannot be embedded into Isom(W_1(R)).","lang":"eng"}],"issue":"8","title":"Isometric study of Wasserstein spaces - the real line","status":"public","ddc":["515"],"intvolume":" 373","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"7389","oa_version":"Preprint"},{"ec_funded":1,"article_number":"108507","volume":278,"date_updated":"2023-08-18T06:36:10Z","date_created":"2020-02-23T23:00:36Z","author":[{"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","orcid":"0000-0002-4821-3297","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger","first_name":"Torben H"},{"full_name":"Nemish, Yuriy","id":"4D902E6A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7327-856X","first_name":"Yuriy","last_name":"Nemish"}],"publisher":"Elsevier","department":[{"_id":"LaEr"}],"publication_status":"published","acknowledgement":"The authors are grateful to Oskari Ajanki for his invaluable help at the initial stage of this project, to Serban Belinschi for useful discussions, to Alexander Tikhomirov for calling our attention to the model example in Section 6.2 and to the anonymous referee for suggesting to simplify certain proofs. Erdös: Partially funded by ERC Advanced Grant RANMAT No. 338804\r\n","year":"2020","publication_identifier":{"eissn":["10960783"],"issn":["00221236"]},"month":"07","language":[{"iso":"eng"}],"doi":"10.1016/j.jfa.2020.108507","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"quality_controlled":"1","isi":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1804.11340"}],"external_id":{"isi":["000522798900001"],"arxiv":["1804.11340"]},"oa":1,"issue":"12","abstract":[{"text":"We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.","lang":"eng"}],"type":"journal_article","oa_version":"Preprint","intvolume":" 278","status":"public","title":"Local laws for polynomials of Wigner matrices","_id":"7512","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","day":"01","scopus_import":"1","date_published":"2020-07-01T00:00:00Z","article_type":"original","citation":{"chicago":"Erdös, László, Torben H Krüger, and Yuriy Nemish. “Local Laws for Polynomials of Wigner Matrices.” Journal of Functional Analysis. Elsevier, 2020. https://doi.org/10.1016/j.jfa.2020.108507.","mla":"Erdös, László, et al. “Local Laws for Polynomials of Wigner Matrices.” Journal of Functional Analysis, vol. 278, no. 12, 108507, Elsevier, 2020, doi:10.1016/j.jfa.2020.108507.","short":"L. Erdös, T.H. Krüger, Y. Nemish, Journal of Functional Analysis 278 (2020).","ista":"Erdös L, Krüger TH, Nemish Y. 2020. Local laws for polynomials of Wigner matrices. Journal of Functional Analysis. 278(12), 108507.","apa":"Erdös, L., Krüger, T. H., & Nemish, Y. (2020). Local laws for polynomials of Wigner matrices. Journal of Functional Analysis. Elsevier. https://doi.org/10.1016/j.jfa.2020.108507","ieee":"L. Erdös, T. H. Krüger, and Y. Nemish, “Local laws for polynomials of Wigner matrices,” Journal of Functional Analysis, vol. 278, no. 12. Elsevier, 2020.","ama":"Erdös L, Krüger TH, Nemish Y. Local laws for polynomials of Wigner matrices. Journal of Functional Analysis. 2020;278(12). doi:10.1016/j.jfa.2020.108507"},"publication":"Journal of Functional Analysis"}]