--- _id: '6179' 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." alternative_title: - ISTA Thesis article_processing_charge: No author: - 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: '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' 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.' ieee: 'D. J. Schröder, “From Dyson to Pearcey: Universal statistics in random matrix theory,” Institute of Science and Technology Austria, 2019.' ista: 'Schröder DJ. 2019. From Dyson to Pearcey: Universal statistics in random matrix theory. Institute of Science and Technology Austria.' 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.' short: 'D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix Theory, Institute of Science and Technology Austria, 2019.' date_created: 2019-03-28T08:58:59Z date_published: 2019-03-18T00:00:00Z date_updated: 2024-02-22T14:34:33Z day: '18' ddc: - '515' - '519' degree_awarded: PhD department: - _id: LaEr doi: 10.15479/AT:ISTA:th6179 ec_funded: 1 file: - access_level: closed checksum: 6926f66f28079a81c4937e3764be00fc content_type: application/x-gzip creator: dernst date_created: 2019-03-28T08:53:52Z date_updated: 2020-07-14T12:47:21Z file_id: '6180' file_name: 2019_Schroeder_Thesis.tar.gz file_size: 7104482 relation: source_file - access_level: open_access checksum: 7d0ebb8d1207e89768cdd497a5bf80fb content_type: application/pdf creator: dernst date_created: 2019-03-28T08:53:52Z date_updated: 2020-07-14T12:47:21Z file_id: '6181' file_name: 2019_Schroeder_Thesis.pdf file_size: 4228794 relation: main_file file_date_updated: 2020-07-14T12:47:21Z has_accepted_license: '1' language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: '375' project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication_identifier: issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '1144' relation: part_of_dissertation status: public - id: '6186' relation: part_of_dissertation status: public - id: '6185' relation: part_of_dissertation status: public - id: '6182' relation: part_of_dissertation status: public - id: '1012' relation: part_of_dissertation status: public - id: '6184' relation: part_of_dissertation status: public status: public supervisor: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 title: 'From Dyson to Pearcey: Universal statistics in random matrix theory' type: dissertation user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2019' ...