--- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '20322' abstract: - lang: eng text: For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner–Dyson–Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality (Alt et al. in Ann Probab 48(2):963–1001, 2020), while cusp universality was proven only for Wigner-type matrices with independent entries (Cipolloni et al. in Pure Appl Anal 1:615–707, 2019; Erdős et al. in Commun. Math. Phys. 378:1203–1278, 2018). As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus we also provide a significantly simplified alternative proof of the local eigenvalue universality in the previously studied bulk (Erdős et al. in Forum Math. Sigma 7:E8, 2019) and edge (Alt et al. in Ann Probab 48(2):963–1001, 2020) regimes. acknowledgement: We thank Giorgio Cipolloni for many productive discussions and the anonymous referees for several useful suggestions and spotting some typos. Open access funding provided by Institute of Science and Technology (IST Austria). article_number: '253' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Erdös L, Henheik SJ, Riabov V. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 2025;406(10). doi:10.1007/s00220-025-05417-z apa: Erdös, L., Henheik, S. J., & Riabov, V. (2025). Cusp universality for correlated random matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-025-05417-z chicago: Erdös, László, Sven Joscha Henheik, and Volodymyr Riabov. “Cusp Universality for Correlated Random Matrices.” Communications in Mathematical Physics. Springer Nature, 2025. https://doi.org/10.1007/s00220-025-05417-z. ieee: L. Erdös, S. J. Henheik, and V. Riabov, “Cusp universality for correlated random matrices,” Communications in Mathematical Physics, vol. 406, no. 10. Springer Nature, 2025. ista: Erdös L, Henheik SJ, Riabov V. 2025. Cusp universality for correlated random matrices. Communications in Mathematical Physics. 406(10), 253. mla: Erdös, László, et al. “Cusp Universality for Correlated Random Matrices.” Communications in Mathematical Physics, vol. 406, no. 10, 253, Springer Nature, 2025, doi:10.1007/s00220-025-05417-z. short: L. Erdös, S.J. Henheik, V. Riabov, Communications in Mathematical Physics 406 (2025). corr_author: '1' date_created: 2025-09-10T05:38:17Z date_published: 2025-09-01T00:00:00Z date_updated: 2026-04-07T12:32:19Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s00220-025-05417-z external_id: arxiv: - '2410.06813' isi: - '001565019000005' file: - access_level: open_access checksum: abd32af7b8ca6dc5b9080823a433986b content_type: application/pdf creator: dernst date_created: 2025-09-10T07:48:21Z date_updated: 2025-09-10T07:48:21Z file_id: '20336' file_name: 2025_CommMathPhysics_Erdoes.pdf file_size: 1465827 relation: main_file success: 1 file_date_updated: 2025-09-10T07:48:21Z has_accepted_license: '1' intvolume: ' 406' isi: 1 issue: '10' language: - iso: eng month: '09' oa: 1 oa_version: Published Version publication: Communications in Mathematical Physics publication_identifier: eissn: - 1432-0916 issn: - 0010-3616 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '19547' relation: earlier_version status: public - id: '20575' relation: dissertation_contains status: public scopus_import: '1' status: public title: Cusp universality for correlated 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 406 year: '2025' ... --- OA_place: repository _id: '20576' abstract: - lang: eng text: We prove that a very general class of $N\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the eigenfunctions are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics, and quantum unique ergodicity holds for general diagonal observables with an optimal convergence rate. Our results are valid for general variance profiles, arbitrary single entry distributions, in both real-symmetric and complex-Hermitian symmetry classes. In particular, our work substantially generalizes the recent breakthrough result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian Gaussian block band matrix. The main technical input is the optimal multi-resolvent local laws -- both in the averaged and fully isotropic form. We also generalize the $\sqrtη$-rule from [arXiv:2012.13215] to exploit the additional effect of traceless observables. Our analysis is based on the zigzag strategy, complemented with a new global-scale estimate derived using the static version of the master inequalities, while the zig-step and the a priori estimates on the deterministic approximations are proven dynamically. acknowledgement: " Supported by the ERC\r\nAdvanced Grant ”RMTBeyond” No. 101020331." 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 - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Erdös L, Riabov V. The zigzag strategy for random band matrices. arXiv. doi:10.48550/ARXIV.2506.06441 apa: Erdös, L., & Riabov, V. (n.d.). The zigzag strategy for random band matrices. arXiv. https://doi.org/10.48550/ARXIV.2506.06441 chicago: Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2506.06441. ieee: L. Erdös and V. Riabov, “The zigzag strategy for random band matrices,” arXiv. . ista: Erdös L, Riabov V. The zigzag strategy for random band matrices. arXiv, 10.48550/ARXIV.2506.06441. mla: Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.” ArXiv, doi:10.48550/ARXIV.2506.06441. short: L. Erdös, V. Riabov, ArXiv (n.d.). corr_author: '1' date_created: 2025-10-29T19:09:03Z date_published: 2025-06-06T00:00:00Z date_updated: 2026-04-07T12:32:19Z day: '06' department: - _id: GradSch - _id: LaEr doi: 10.48550/ARXIV.2506.06441 ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2506.06441 month: '06' oa: 1 oa_version: Preprint project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: arXiv publication_status: draft related_material: record: - id: '20575' relation: dissertation_contains status: public status: public title: The zigzag strategy for random band matrices type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ... --- OA_place: publisher _id: '20575' abstract: - lang: eng text: "This thesis deals with eigenvalue and eigenvector universality results for random matrix ensembles equipped with non-trivial spatial structure. We consider both mean-field models with a general variance profile (Wigner-type matrices) and correlation structure (correlated matrices) among the entries, as well as non-mean-field random band matrices with bandwidth W >> N^(1/2).\r\n\r\nTo extract the universal properties of random matrix spectra and eigenvectors, we obtain concentration estimates for their resolvent, the local laws, which generalize the celebrated Wigner semicircle law for a broad class of random matrices to much finer spectral scales. The local laws hold for both a single resolvent as well as for products of multiple resolvents, known as resolvent chains, and express the remarkable approximately-deterministic behavior of these objects down to the microscopic scale.\r\n\r\nOur primary tool for establishing the local laws is the dynamical Zigzag strategy, which we develop in the setting of spatially-inhomogeneous random matrices. Our proof method systematically addresses the challenges arising from non-trivial spatial structures and is robust to all types of singularities in the spectrum, as we demonstrate in the correlated setting. Furthermore, we incorporate the analysis of the deterministic resolvent chain approximations into the dynamical framework of the Zigzag strategy, synthesizing a unified toolkit for establishing multi-resolvent local laws.\r\n\r\nUsing these methods, we prove complete eigenvector delocalization, the Eigenstate Thermalization Hypothesis, and Wigner-Dyson universality in the bulk for random band matrices down to the optimal bandwidth W >> N^(1/2). For mean-field ensembles, we establish universality of local eigenvalue statistics at the cups for random matrices with correlated entries, and the Eigenstate Thermalization Hypothesis for Wigner-type matrices in the bulk of the spectrum.\r\n\r\nFinally, this thesis also contains other applications of the multi-resolvent local laws to spatially-inhomogeneous random matrices, obtained prior to the development of the Zigzag strategy. In particular, we provide a complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-type matrices in all spectral regimes, including the novel cusps, and rigorously establish the prethermalization phenomenon for deformed Wigner matrices.\r\n\r\nThe main body of this thesis consists of seven research papers (listed on page xi), each presented in a separate chapter with its own introduction and all relevant context, suitable to be read independently. We ask the reader’s indulgence for the repetitions in the historical overviews and other minor redundancies that remain among the chapters as a result. The overall Introduction, preceding the chapters, provides a condensed, informal summary of the main ideas and concepts at the core of these works.\r\n" acknowledgement: "The work comprising this thesis was supported by the ERC Advanced Grant \"RMTBeyond\"\r\nNo.101020331 awarded to my advisor." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Riabov V. Universality in random matrices with spatial structure. 2025. doi:10.15479/AT-ISTA-20575 apa: Riabov, V. (2025). Universality in random matrices with spatial structure. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-20575 chicago: Riabov, Volodymyr. “Universality in Random Matrices with Spatial Structure.” Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/AT-ISTA-20575. ieee: V. Riabov, “Universality in random matrices with spatial structure,” Institute of Science and Technology Austria, 2025. ista: Riabov V. 2025. Universality in random matrices with spatial structure. Institute of Science and Technology Austria. mla: Riabov, Volodymyr. Universality in Random Matrices with Spatial Structure. Institute of Science and Technology Austria, 2025, doi:10.15479/AT-ISTA-20575. short: V. Riabov, Universality in Random Matrices with Spatial Structure, Institute of Science and Technology Austria, 2025. corr_author: '1' date_created: 2025-10-29T19:12:24Z date_published: 2025-11-03T00:00:00Z date_updated: 2026-04-07T12:32:20Z day: '3' ddc: - '515' - '519' degree_awarded: PhD department: - _id: GradSch - _id: LaEr doi: 10.15479/AT-ISTA-20575 ec_funded: 1 file: - access_level: open_access checksum: 6a0487b2b66bb35d44b394756d44b8b4 content_type: application/pdf creator: vriabov date_created: 2025-10-29T18:53:59Z date_updated: 2025-10-29T18:53:59Z file_id: '20577' file_name: riabov_thesis-pdfa.pdf file_size: 7536583 relation: main_file success: 1 - access_level: closed checksum: 224efda6bf9864d296a1e5e0124c1e8f content_type: application/x-zip-compressed creator: vriabov date_created: 2025-10-29T18:54:53Z date_updated: 2025-10-29T18:54:53Z file_id: '20578' file_name: manuscript.zip file_size: 17841612 relation: source_file file_date_updated: 2025-10-29T18:54:53Z has_accepted_license: '1' language: - iso: eng month: '11' oa: 1 oa_version: Published Version page: '436' project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication_identifier: isbn: - 978-3-99078-064-0 issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '20322' relation: part_of_dissertation status: public - id: '18764' relation: part_of_dissertation status: public - id: '13317' relation: part_of_dissertation status: public - id: '19368' relation: part_of_dissertation status: deleted - id: '18554' relation: part_of_dissertation status: public - id: '20576' relation: part_of_dissertation status: public - id: '17174' relation: part_of_dissertation status: public - id: '19547' relation: part_of_dissertation status: public - id: '19598' 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: Universality in random matrices with spatial structure 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: dissertation user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd year: '2025' ... --- OA_place: publisher OA_type: hybrid PlanS_conform: '1' _id: '19598' abstract: - lang: eng text: We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics in the vicinity of the cusp-like singularities of the limiting spectral density for Wigner-type random matrices. Prior to this work, the linear eigenvalue statistics at the cusp-like singularities were not studied in any ensemble. Our analysis covers not only the exact cusps but the entire transitionary regime from the square-root singularity at a regular edge through the sharp cusp to the bulk. We identify a new one-parameter family of functionals that govern the limiting bias and variance, continuously interpolating between the previously known formulas in the bulk and at a regular edge. Since cusps are the only possible singularities besides the regular edges, our result gives a complete description of the linear eigenvalue statistics in all regimes. acknowledgement: I would like to express my gratitude to László Erdős for his careful guidance and supervision of my work. I am also thankful to Jana Reker and Joscha Henheik for many helpful discussions. Open access funding provided by Institute of Science and Technology (IST Austria). article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Riabov V. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. 2025;193:1183-1237. doi:10.1007/s00440-025-01373-w apa: Riabov, V. (2025). Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-025-01373-w chicago: Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” Probability Theory and Related Fields. Springer Nature, 2025. https://doi.org/10.1007/s00440-025-01373-w. ieee: V. Riabov, “Linear Eigenvalue statistics at the cusp,” Probability Theory and Related Fields, vol. 193. Springer Nature, pp. 1183–1237, 2025. ista: Riabov V. 2025. Linear Eigenvalue statistics at the cusp. Probability Theory and Related Fields. 193, 1183–1237. mla: Riabov, Volodymyr. “Linear Eigenvalue Statistics at the Cusp.” Probability Theory and Related Fields, vol. 193, Springer Nature, 2025, pp. 1183–237, doi:10.1007/s00440-025-01373-w. short: V. Riabov, Probability Theory and Related Fields 193 (2025) 1183–1237. corr_author: '1' date_created: 2025-04-20T22:01:28Z date_published: 2025-12-01T00:00:00Z date_updated: 2026-04-07T12:32:19Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s00440-025-01373-w external_id: arxiv: - '2307.07432' isi: - '001466997300001' file: - access_level: open_access checksum: 700229b280725c0d6aad0d71362cce5f content_type: application/pdf creator: dernst date_created: 2025-12-30T13:10:05Z date_updated: 2025-12-30T13:10:05Z file_id: '20916' file_name: 2025_ProbTheoryRelatFields_Riabov.pdf file_size: 919213 relation: main_file success: 1 file_date_updated: 2025-12-30T13:10:05Z has_accepted_license: '1' intvolume: ' 193' isi: 1 language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 1183-1237 publication: Probability Theory and Related Fields publication_identifier: eissn: - 1432-2064 issn: - 0178-8051 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '20575' relation: dissertation_contains status: public scopus_import: '1' status: public title: Linear Eigenvalue statistics at the cusp 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: 193 year: '2025' ... --- OA_place: publisher OA_type: hybrid _id: '18112' abstract: - lang: eng text: 'It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.' acknowledgement: I am very grateful to Vadim Kaloshin for suggesting the topic, his guidance during this project, and many helpful comments on an earlier version of the manuscript. Moreover, I would like to thank Comlan Edmond Koudjinan and Volodymyr Riabov for interesting discussions. Partial financial support by the ERC Advanced Grant ‘RMTBeyond’ No. 101020331 is gratefully acknowledged. This project received funding from the European Research Council (ERC) ERC Grant No. 885707. article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X citation: ama: Henheik SJ. Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. 2025;45(2):467-503. doi:10.1017/etds.2024.48 apa: Henheik, S. J. (2025). Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. Cambridge University Press. https://doi.org/10.1017/etds.2024.48 chicago: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the Torus.” Ergodic Theory and Dynamical Systems. Cambridge University Press, 2025. https://doi.org/10.1017/etds.2024.48. ieee: S. J. Henheik, “Deformational rigidity of integrable metrics on the torus,” Ergodic Theory and Dynamical Systems, vol. 45, no. 2. Cambridge University Press, pp. 467–503, 2025. ista: Henheik SJ. 2025. Deformational rigidity of integrable metrics on the torus. Ergodic Theory and Dynamical Systems. 45(2), 467–503. mla: Henheik, Sven Joscha. “Deformational Rigidity of Integrable Metrics on the Torus.” Ergodic Theory and Dynamical Systems, vol. 45, no. 2, Cambridge University Press, 2025, pp. 467–503, doi:10.1017/etds.2024.48. short: S.J. Henheik, Ergodic Theory and Dynamical Systems 45 (2025) 467–503. corr_author: '1' date_created: 2024-09-22T22:01:43Z date_published: 2025-02-01T00:00:00Z date_updated: 2026-04-07T12:37:10Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1017/etds.2024.48 ec_funded: 1 external_id: isi: - '001308182000001' file: - access_level: open_access checksum: 650fe115d998fe0ac3a8d0c7519447c8 content_type: application/pdf creator: dernst date_created: 2025-01-13T08:51:40Z date_updated: 2025-01-13T08:51:40Z file_id: '18828' file_name: 2025_ErgodicTheory_Henheik.pdf file_size: 659100 relation: main_file success: 1 file_date_updated: 2025-01-13T08:51:40Z has_accepted_license: '1' intvolume: ' 45' isi: 1 issue: '2' language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: 467-503 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta - _id: 9B8B92DE-BA93-11EA-9121-9846C619BF3A call_identifier: H2020 grant_number: '885707' name: Spectral rigidity and integrability for billiards and geodesic flows publication: Ergodic Theory and Dynamical Systems publication_identifier: eissn: - 1469-4417 issn: - 0143-3857 publication_status: published publisher: Cambridge University Press quality_controlled: '1' related_material: record: - id: '19540' relation: dissertation_contains status: public scopus_import: '1' status: public title: Deformational rigidity of integrable metrics on the torus 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: 45 year: '2025' ... --- OA_place: publisher OA_type: hybrid _id: '19001' abstract: - lang: eng text: We consider two Hamiltonians that are close to each other, H1≈H2, and analyze the time-decay of the corresponding Loschmidt echo M(t):=|⟨ψ0,eitH2e−itH1ψ0⟩|2 that expresses the effect of an imperfect time reversal on the initial state ψ0. Our model Hamiltonians are deformed Wigner matrices that do not share a common eigenbasis. The main tools for our results are two-resolvent laws for such H1 and H2. acknowledgement: We thank Giorgio Cipolloni for helpful discussions in a closely related joint project. Open access funding provided by Institute of Science and Technology (IST Austria). All authors were supported by the ERC Advanced Grant “RMTBeyond” No. 101020331. article_number: '14' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Oleksii full_name: Kolupaiev, Oleksii id: 149b70d4-896a-11ed-bdf8-8c63fd44ca61 last_name: Kolupaiev orcid: 0000-0003-1491-4623 citation: ama: Erdös L, Henheik SJ, Kolupaiev O. Loschmidt echo for deformed Wigner matrices. Letters in Mathematical Physics. 2025;115. doi:10.1007/s11005-025-01904-5 apa: Erdös, L., Henheik, S. J., & Kolupaiev, O. (2025). Loschmidt echo for deformed Wigner matrices. Letters in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s11005-025-01904-5 chicago: Erdös, László, Sven Joscha Henheik, and Oleksii Kolupaiev. “Loschmidt Echo for Deformed Wigner Matrices.” Letters in Mathematical Physics. Springer Nature, 2025. https://doi.org/10.1007/s11005-025-01904-5. ieee: L. Erdös, S. J. Henheik, and O. Kolupaiev, “Loschmidt echo for deformed Wigner matrices,” Letters in Mathematical Physics, vol. 115. Springer Nature, 2025. ista: Erdös L, Henheik SJ, Kolupaiev O. 2025. Loschmidt echo for deformed Wigner matrices. Letters in Mathematical Physics. 115, 14. mla: Erdös, László, et al. “Loschmidt Echo for Deformed Wigner Matrices.” Letters in Mathematical Physics, vol. 115, 14, Springer Nature, 2025, doi:10.1007/s11005-025-01904-5. short: L. Erdös, S.J. Henheik, O. Kolupaiev, Letters in Mathematical Physics 115 (2025). corr_author: '1' date_created: 2025-02-05T06:48:29Z date_published: 2025-01-30T00:00:00Z date_updated: 2026-04-07T12:37:10Z day: '30' ddc: - '510' department: - _id: LaEr doi: 10.1007/s11005-025-01904-5 ec_funded: 1 external_id: arxiv: - '2410.08108' isi: - '001409618800002' pmid: - '39896265' file: - access_level: open_access checksum: ee07edf5f85a6f2651926b2f8760af74 content_type: application/pdf creator: dernst date_created: 2025-02-05T07:01:40Z date_updated: 2025-02-05T07:01:40Z file_id: '19004' file_name: 2025_LettersMathPhysics_Erdoes.pdf file_size: 828335 relation: main_file success: 1 file_date_updated: 2025-02-05T07:01:40Z has_accepted_license: '1' intvolume: ' 115' isi: 1 language: - iso: eng month: '01' oa: 1 oa_version: Published Version pmid: 1 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Letters in Mathematical Physics publication_identifier: issn: - 1573-0530 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '19540' relation: dissertation_contains status: public scopus_import: '1' status: public title: Loschmidt echo for deformed 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: 115 year: '2025' ... --- OA_place: publisher OA_type: hybrid _id: '18764' abstract: - lang: eng text: We prove that a class of weakly perturbed Hamiltonians of the form H_λ= H_0 + λW, with W being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by H_λ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order λ^{-2}. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix H_λ. acknowledgement: "All authors were supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nJ.R. was additionally supported by the ERC Advanced Grant “LDRaM” No. 884584.\r\nWe thank Peter Reimann and Lennart Dabelow for helpful comments. Open access funding provided by Institute of Science and Technology (IST Austria)." article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Jana full_name: Reker, Jana id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9 last_name: Reker - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner matrices. Annales Henri Poincare. 2025;26:1991-2033. doi:10.1007/s00023-024-01518-y apa: Erdös, L., Henheik, S. J., Reker, J., & Riabov, V. (2025). Prethermalization for deformed Wigner matrices. Annales Henri Poincare. Springer Nature. https://doi.org/10.1007/s00023-024-01518-y chicago: Erdös, László, Sven Joscha Henheik, Jana Reker, and Volodymyr Riabov. “Prethermalization for Deformed Wigner Matrices.” Annales Henri Poincare. Springer Nature, 2025. https://doi.org/10.1007/s00023-024-01518-y. ieee: L. Erdös, S. J. Henheik, J. Reker, and V. Riabov, “Prethermalization for deformed Wigner matrices,” Annales Henri Poincare, vol. 26. Springer Nature, pp. 1991–2033, 2025. ista: Erdös L, Henheik SJ, Reker J, Riabov V. 2025. Prethermalization for deformed Wigner matrices. Annales Henri Poincare. 26, 1991–2033. mla: Erdös, László, et al. “Prethermalization for Deformed Wigner Matrices.” Annales Henri Poincare, vol. 26, Springer Nature, 2025, pp. 1991–2033, doi:10.1007/s00023-024-01518-y. short: L. Erdös, S.J. Henheik, J. Reker, V. Riabov, Annales Henri Poincare 26 (2025) 1991–2033. corr_author: '1' date_created: 2025-01-05T23:01:59Z date_published: 2025-06-01T00:00:00Z date_updated: 2026-04-07T12:37:11Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s00023-024-01518-y ec_funded: 1 external_id: arxiv: - '2310.06677' isi: - '001385326500001' file: - access_level: open_access checksum: 49e6a934db540206f7eaa0c798553ded content_type: application/pdf creator: dernst date_created: 2025-06-25T05:38:34Z date_updated: 2025-06-25T05:38:34Z file_id: '19895' file_name: 2025_AnnalesHenriPoincare_Erdoes.pdf file_size: 977773 relation: main_file success: 1 file_date_updated: 2025-06-25T05:38:34Z has_accepted_license: '1' intvolume: ' 26' isi: 1 language: - iso: eng month: '06' oa: 1 oa_version: Published Version page: 1991-2033 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Annales Henri Poincare publication_identifier: issn: - 1424-0637 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '17174' relation: earlier_version status: public - id: '20575' relation: dissertation_contains status: public - id: '19540' relation: dissertation_contains status: public scopus_import: '1' status: public title: Prethermalization for deformed 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 26 year: '2025' ... --- OA_place: repository _id: '19552' abstract: - lang: eng text: "Particle creation terms in quantum Hamiltonians are usually ultraviolet\r\ndivergent and thus mathematically ill defined. A rather novel way of solving\r\nthis problem is based on imposing so-called interior-boundary conditions on the\r\nwave function. Previous papers showed that this approach works in the\r\nnon-relativistic regime, but particle creation is mostly relevant in the\r\nrelativistic case after all. In flat relativistic space-time (that is,\r\nneglecting gravity), the approach was previously found to work only for certain\r\nsomewhat artificial cases. Here, as a way of taking gravity into account, we\r\nconsider curved space-time, specifically the super-critical\r\nReissner-Nordstr\\\"om space-time, which features a naked timelike singularity.\r\nWe find that the interior-boundary approach works fully in this setting; in\r\nparticular, we prove rigorously the existence of well-defined, self-adjoint\r\nHamiltonians with particle creation at the singularity, based on\r\ninterior-boundary conditions. We also non-rigorously analyze the asymptotic\r\nbehavior of the Bohmian trajectories and construct the corresponding Bohm-Bell\r\nprocess of particle creation, motion, and annihilation. The upshot is that in\r\nquantum physics, a naked space-time singularity need not lead to a breakdown of\r\nphysical laws, but on the contrary allows for boundary conditions governing\r\nwhat comes out of the singularity and thereby removing the ultraviolet\r\ndivergence." acknowledgement: "JH gratefully acknowledges partial financial support by the ERC Advanced\r\nGrant “RMTBeyond” No. 101020331." article_processing_charge: No arxiv: 1 author: - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Bipul full_name: Poudyal, Bipul last_name: Poudyal - first_name: Roderich full_name: Tumulka, Roderich last_name: Tumulka citation: ama: Henheik SJ, Poudyal B, Tumulka R. How a space-time singularity helps remove the ultraviolet divergence problem. arXiv. doi:10.48550/arXiv.2409.00677 apa: Henheik, S. J., Poudyal, B., & Tumulka, R. (n.d.). How a space-time singularity helps remove the ultraviolet divergence problem. arXiv. https://doi.org/10.48550/arXiv.2409.00677 chicago: Henheik, Sven Joscha, Bipul Poudyal, and Roderich Tumulka. “How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2409.00677. ieee: S. J. Henheik, B. Poudyal, and R. Tumulka, “How a space-time singularity helps remove the ultraviolet divergence problem,” arXiv. . ista: Henheik SJ, Poudyal B, Tumulka R. How a space-time singularity helps remove the ultraviolet divergence problem. arXiv, 10.48550/arXiv.2409.00677. mla: Henheik, Sven Joscha, et al. “How a Space-Time Singularity Helps Remove the Ultraviolet Divergence Problem.” ArXiv, doi:10.48550/arXiv.2409.00677. short: S.J. Henheik, B. Poudyal, R. Tumulka, ArXiv (n.d.). corr_author: '1' date_created: 2025-04-11T12:07:25Z date_published: 2025-02-28T00:00:00Z date_updated: 2026-04-07T12:37:11Z day: '28' department: - _id: LaEr doi: 10.48550/arXiv.2409.00677 ec_funded: 1 external_id: arxiv: - '2409.00677' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2409.00677 month: '02' oa: 1 oa_version: Preprint project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: arXiv publication_status: draft related_material: record: - id: '19540' relation: dissertation_contains status: public status: public title: How a space-time singularity helps remove the ultraviolet divergence problem type: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ... --- OA_place: repository _id: '19546' abstract: - lang: eng text: "We study the sensitivity of the eigenvectors of random matrices, showing that\r\neven small perturbations make the eigenvectors almost orthogonal. More\r\nprecisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show\r\nthat their bulk eigenvectors become asymptotically orthogonal as soon as\r\n$\\mathrm{Tr}(D_1-D_2)^2\\gg 1$, or their respective energies are separated on a\r\nscale much bigger than the local eigenvalue spacing. Furthermore, we show that\r\nquadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic\r\nmatrix $A\\in\\mathbf{C}^{N\\times N}$ in a specific subspace of codimension one\r\nare of size $N^{-1/2}$. This proves a generalization of the Eigenstate\r\nThermalization Hypothesis to eigenvectors belonging to two different spectral\r\nfamilies." acknowledgement: Supported by the ERC Advanced Grant “RMTBeyond” No. 101020331. article_processing_charge: No arxiv: 1 author: - first_name: Giorgio full_name: Cipolloni, Giorgio id: 42198EFA-F248-11E8-B48F-1D18A9856A87 last_name: Cipolloni orcid: 0000-0002-4901-7992 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Oleksii full_name: Kolupaiev, Oleksii id: 149b70d4-896a-11ed-bdf8-8c63fd44ca61 last_name: Kolupaiev orcid: 0000-0003-1491-4623 citation: ama: Cipolloni G, Erdös L, Henheik SJ, Kolupaiev O. Eigenvector decorrelation for random matrices. arXiv. doi:10.48550/arXiv.2410.10718 apa: Cipolloni, G., Erdös, L., Henheik, S. J., & Kolupaiev, O. (n.d.). Eigenvector decorrelation for random matrices. arXiv. https://doi.org/10.48550/arXiv.2410.10718 chicago: Cipolloni, Giorgio, László Erdös, Sven Joscha Henheik, and Oleksii Kolupaiev. “Eigenvector Decorrelation for Random Matrices.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2410.10718. ieee: G. Cipolloni, L. Erdös, S. J. Henheik, and O. Kolupaiev, “Eigenvector decorrelation for random matrices,” arXiv. . ista: Cipolloni G, Erdös L, Henheik SJ, Kolupaiev O. Eigenvector decorrelation for random matrices. arXiv, 10.48550/arXiv.2410.10718. mla: Cipolloni, Giorgio, et al. “Eigenvector Decorrelation for Random Matrices.” ArXiv, doi:10.48550/arXiv.2410.10718. short: G. Cipolloni, L. Erdös, S.J. Henheik, O. Kolupaiev, ArXiv (n.d.). corr_author: '1' date_created: 2025-04-11T08:34:49Z date_published: 2025-01-30T00:00:00Z date_updated: 2026-04-07T12:37:11Z day: '30' department: - _id: LaEr doi: 10.48550/arXiv.2410.10718 ec_funded: 1 external_id: arxiv: - '2410.10718' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2410.10718 month: '01' oa: 1 oa_version: Preprint project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: arXiv publication_status: draft related_material: record: - id: '19540' relation: dissertation_contains status: public status: public title: Eigenvector decorrelation for 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: preprint user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2025' ... --- OA_place: publisher _id: '19540' abstract: - lang: eng text: "This thesis deals with several different models for complex quantum mechanical systems and is structured in three main parts. \r\n\t\r\nIn Part I, we study mean field random matrices as models for quantum Hamiltonians. Our focus lies on proving concentration estimates for resolvents of random matrices, so-called local laws, mostly in the setting of multiple resolvents. These estimates have profound consequences for eigenvector overlaps and thermalization problems. More concretely, we obtain, e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector overlaps, and prethermalization for deformed Wigner matrices.\tIn order to prove our novel multi-resolvent local laws, we develop and devise two main methods, the static Psi-method and the dynamical Zigzag strategy. \r\n\t\r\nIn Part II, we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic theory of superconductivity. We focus on asymptotic formulas for the characteristic critical temperature and energy gap of a superconductor and prove universality of their ratio in various physical regimes. Additionally, we investigate multi-band superconductors and show that inter-band coupling effects can only enhance the critical temperature. \r\n\t\r\nIn Part III, we study quantum lattice systems. On the one hand, we show a strong version of the local-perturbations-perturb-locally (LPPL) principle for the ground state of weakly interacting quantum spin systems with a uniform on-site gap. On the other hand, we introduce a notion of a local gap and rigorously justify response theory and the Kubo formula under the weakened assumption of a local gap. \r\n\t\r\nAdditionally, we discuss two classes of problems which do not fit into the three main parts of the thesis. These are deformational rigidity of Liouville metrics on the torus and relativistic toy models of particle creation via interior-boundary-conditions (IBCs). " alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X citation: ama: 'Henheik SJ. Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems. 2025. doi:10.15479/AT-ISTA-19540' apa: 'Henheik, S. J. (2025). Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-19540' chicago: 'Henheik, Sven Joscha. “Modeling Complex Quantum Systems : Random Matrices, BCS Theory, and Quantum Lattice Systems.” Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/AT-ISTA-19540.' ieee: 'S. J. Henheik, “Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems,” Institute of Science and Technology Austria, 2025.' ista: 'Henheik SJ. 2025. Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems. Institute of Science and Technology Austria.' mla: 'Henheik, Sven Joscha. Modeling Complex Quantum Systems : Random Matrices, BCS Theory, and Quantum Lattice Systems. Institute of Science and Technology Austria, 2025, doi:10.15479/AT-ISTA-19540.' short: 'S.J. Henheik, Modeling Complex Quantum Systems : Random Matrices, BCS Theory, and Quantum Lattice Systems, Institute of Science and Technology Austria, 2025.' corr_author: '1' date_created: 2025-04-10T21:21:18Z date_published: 2025-04-10T00:00:00Z date_updated: 2026-04-07T12:37:12Z day: '10' ddc: - '519' degree_awarded: PhD department: - _id: GradSch - _id: LaEr doi: 10.15479/AT-ISTA-19540 ec_funded: 1 file: - access_level: closed checksum: b8477ae5578436c72c3bb4193ad34ac5 content_type: application/zip creator: shenheik date_created: 2025-04-10T21:14:18Z date_updated: 2025-04-10T21:14:18Z file_id: '19542' file_name: Henheik-Thesis_source_final.zip file_size: 4107587 relation: source_file - access_level: open_access checksum: e9fc0ea12ec46c9f71110c33217c4140 content_type: application/pdf creator: shenheik date_created: 2025-04-11T13:16:05Z date_updated: 2025-04-11T13:16:05Z file_id: '19553' file_name: Henheik-Thesis-pdfa_FINAL.pdf file_size: 9999492 relation: main_file success: 1 - access_level: closed checksum: f94580f86c785e7108eb116cd189e225 content_type: application/pdf creator: cchlebak date_created: 2025-04-23T14:10:27Z date_updated: 2025-04-23T14:10:27Z file_id: '19615' file_name: Henheik-Thesis-Volume1_print.pdf file_size: 13276442 relation: other - access_level: closed checksum: b927ead3c78020ffb32918911deedb74 content_type: application/pdf creator: cchlebak date_created: 2025-04-23T14:11:05Z date_updated: 2025-04-23T14:11:05Z file_id: '19616' file_name: Henheik-Thesis-Volume2_print.pdf file_size: 7628767 relation: other file_date_updated: 2025-04-23T14:11:05Z has_accepted_license: '1' language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: '720' project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication_identifier: isbn: - 978-3-99078-057-2 issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology Austria related_material: record: - id: '14343' relation: part_of_dissertation status: public - id: '18656' relation: part_of_dissertation status: public - id: '13317' relation: part_of_dissertation status: public - id: '11732' relation: part_of_dissertation status: public - id: '12184' relation: part_of_dissertation status: public - id: '14421' relation: part_of_dissertation status: public - id: '10623' relation: part_of_dissertation status: public - id: '18112' relation: part_of_dissertation status: public - id: '19001' relation: part_of_dissertation status: public - id: '10642' relation: part_of_dissertation status: public - id: '19545' relation: part_of_dissertation status: public - id: '19546' relation: part_of_dissertation status: public - id: '19550' relation: part_of_dissertation status: public - id: '19551' relation: part_of_dissertation status: public - id: '19552' relation: part_of_dissertation status: public - id: '14542' relation: part_of_dissertation status: public - id: '17049' relation: part_of_dissertation status: public - id: '18764' relation: part_of_dissertation status: public - id: '19547' relation: part_of_dissertation status: public - id: '19548' 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: 'Modeling complex quantum systems : Random matrices, BCS theory, and quantum lattice systems' 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: dissertation user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd year: '2025' ... --- DOAJ_listed: '1' OA_place: publisher OA_type: gold _id: '19548' abstract: - lang: eng text: "We consider the BCS energy gap „.T / (essentially given by „.T / \x19 \x81.T; p\x16/,\r\nthe BCS order parameter) at all temperatures 0 \x14 T \x14 Tc up to the critical one, Tc, and show\r\nthat, in the limit of weak coupling, the ratio „.T /=Tc is given by a universal function of the relative temperature T =Tc. On the one hand, this recovers a recent result by Langmann and Triola\r\n[Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors for temperatures bounded uniformly away from Tc. On the other hand, our result lifts these\r\nrestrictions, as we consider arbitrary spatial dimensions d 2 ¹1; 2; 3º, discuss superconductors\r\nwith non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of\r\ntemperatures close to Tc.\r\n\r\n​\r\n ." acknowledgement: "We thank Andreas Deuchert, Christian Hainzl, Edwin Langmann, Marius Lemm, Robert Seiringer, and Jan Philip Solovej for helpful discussions,\r\nand Edwin Langmann and Robert Seiringer for valuable comments on an earlier version of the manuscript.\r\nFunding. Joscha Henheik gratefully acknowledges partial financial support by the\r\nERC Advanced Grant “RMTBeyond” No. 101020331. Asbjørn Bækgaard Lauritsen\r\ngratefully acknowledges partial financial support by the Austrian Science Fund (FWF)\r\nthrough grant DOI 10.55776/I6427 (as part of the SFB/TRR 352).\r\n" article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X - first_name: Asbjørn Bækgaard full_name: Lauritsen, Asbjørn Bækgaard id: e1a2682f-dc8d-11ea-abe3-81da9ac728f1 last_name: Lauritsen orcid: 0000-0003-4476-2288 citation: ama: Henheik SJ, Lauritsen AB. Universal behavior of the BCS energy gap. Journal of Spectral Theory. 2025;15(1):305–352. doi:10.4171/JST/540 apa: Henheik, S. J., & Lauritsen, A. B. (2025). Universal behavior of the BCS energy gap. Journal of Spectral Theory. EMS Press. https://doi.org/10.4171/JST/540 chicago: Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior of the BCS Energy Gap.” Journal of Spectral Theory. EMS Press, 2025. https://doi.org/10.4171/JST/540. ieee: S. J. Henheik and A. B. Lauritsen, “Universal behavior of the BCS energy gap,” Journal of Spectral Theory, vol. 15, no. 1. EMS Press, pp. 305–352, 2025. ista: Henheik SJ, Lauritsen AB. 2025. Universal behavior of the BCS energy gap. Journal of Spectral Theory. 15(1), 305–352. mla: Henheik, Sven Joscha, and Asbjørn Bækgaard Lauritsen. “Universal Behavior of the BCS Energy Gap.” Journal of Spectral Theory, vol. 15, no. 1, EMS Press, 2025, pp. 305–352, doi:10.4171/JST/540. short: S.J. Henheik, A.B. Lauritsen, Journal of Spectral Theory 15 (2025) 305–352. corr_author: '1' date_created: 2025-04-11T09:19:28Z date_published: 2025-01-09T00:00:00Z date_updated: 2026-04-07T12:37:11Z day: '09' ddc: - '500' department: - _id: LaEr - _id: RoSe doi: 10.4171/JST/540 ec_funded: 1 external_id: arxiv: - '2312.11310' isi: - '001438931600009' file: - access_level: open_access checksum: f49e06e8dba819f7ad52a202e287ebca content_type: application/pdf creator: cchlebak date_created: 2025-04-11T09:13:31Z date_updated: 2025-04-11T09:13:31Z file_id: '19549' file_name: Henheik_JSpectralTheory_2025.pdf file_size: 779158 relation: main_file success: 1 file_date_updated: 2025-04-11T09:13:31Z has_accepted_license: '1' intvolume: ' 15' isi: 1 issue: '1' language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 305–352 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta - _id: bda63fe5-d553-11ed-ba76-a16e3d2f256b grant_number: I06427 name: Mathematical Challenges in BCS Theory of Superconductivity publication: Journal of Spectral Theory publication_identifier: eissn: - 1664-0403 publication_status: published publisher: EMS Press quality_controlled: '1' related_material: record: - id: '19540' relation: dissertation_contains status: public scopus_import: '1' status: public title: Universal behavior of the BCS energy gap 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: 15 year: '2025' ... --- _id: '13975' abstract: - lang: eng text: "We consider the spectrum of random Laplacian matrices of the form Ln=An−Dn where An\r\n is a real symmetric random matrix and Dn is a diagonal matrix whose entries are equal to the corresponding row sums of An. If An is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution, the empirical spectral measure of Ln is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices An with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math Soc, 370, (2018)]. Our main result shows that the empirical spectral measure of Ln converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which Ln converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure." acknowledgement: "The first author thanks Yizhe Zhu for pointing out reference [30]. We thank David Renfrew for comments on an earlier draft. We thank the anonymous referee for a careful reading and helpful comments.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria)." article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: Andrew J full_name: Campbell, Andrew J id: 582b06a9-1f1c-11ee-b076-82ffce00dde4 last_name: Campbell - first_name: Sean full_name: O’Rourke, Sean last_name: O’Rourke citation: ama: Campbell AJ, O’Rourke S. Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability. 2024;37:933-973. doi:10.1007/s10959-023-01275-4 apa: Campbell, A. J., & O’Rourke, S. (2024). Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability. Springer Nature. https://doi.org/10.1007/s10959-023-01275-4 chicago: Campbell, Andrew J, and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random Laplacian Matrices.” Journal of Theoretical Probability. Springer Nature, 2024. https://doi.org/10.1007/s10959-023-01275-4. ieee: A. J. Campbell and S. O’Rourke, “Spectrum of Lévy–Khintchine random laplacian matrices,” Journal of Theoretical Probability, vol. 37. Springer Nature, pp. 933–973, 2024. ista: Campbell AJ, O’Rourke S. 2024. Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability. 37, 933–973. mla: Campbell, Andrew J., and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random Laplacian Matrices.” Journal of Theoretical Probability, vol. 37, Springer Nature, 2024, pp. 933–73, doi:10.1007/s10959-023-01275-4. short: A.J. Campbell, S. O’Rourke, Journal of Theoretical Probability 37 (2024) 933–973. corr_author: '1' date_created: 2023-08-06T22:01:13Z date_published: 2024-03-01T00:00:00Z date_updated: 2024-07-22T09:41:42Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s10959-023-01275-4 external_id: arxiv: - '2210.07927' isi: - '001038341000001' file: - access_level: open_access checksum: f7793d313104c70422140c5e6494c779 content_type: application/pdf creator: dernst date_created: 2024-07-22T09:41:21Z date_updated: 2024-07-22T09:41:21Z file_id: '17300' file_name: 2024_JourTheorProbab_Campbell.pdf file_size: 555070 relation: main_file success: 1 file_date_updated: 2024-07-22T09:41:21Z has_accepted_license: '1' intvolume: ' 37' isi: 1 language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: 933-973 publication: Journal of Theoretical Probability publication_identifier: eissn: - 1572-9230 issn: - 0894-9840 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Spectrum of Lévy–Khintchine random laplacian 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: 37 year: '2024' ... --- _id: '14408' abstract: - lang: eng text: "We prove that the mesoscopic linear statistics ∑if(na(σi−z0)) of the eigenvalues {σi}i of large n×n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H20-functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0Probability Theory and Related Fields. 2024;188:1131-1182. doi:10.1007/s00440-023-01229-1 apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2024). Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. Springer Nature. https://doi.org/10.1007/s00440-023-01229-1 chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Mesoscopic Central Limit Theorem for Non-Hermitian Random Matrices.” Probability Theory and Related Fields. Springer Nature, 2024. https://doi.org/10.1007/s00440-023-01229-1. ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Mesoscopic central limit theorem for non-Hermitian random matrices,” Probability Theory and Related Fields, vol. 188. Springer Nature, pp. 1131–1182, 2024. ista: Cipolloni G, Erdös L, Schröder DJ. 2024. Mesoscopic central limit theorem for non-Hermitian random matrices. Probability Theory and Related Fields. 188, 1131–1182. mla: Cipolloni, Giorgio, et al. “Mesoscopic Central Limit Theorem for Non-Hermitian Random Matrices.” Probability Theory and Related Fields, vol. 188, Springer Nature, 2024, pp. 1131–82, doi:10.1007/s00440-023-01229-1. short: G. Cipolloni, L. Erdös, D.J. Schröder, Probability Theory and Related Fields 188 (2024) 1131–1182. date_created: 2023-10-08T22:01:17Z date_published: 2024-04-01T00:00:00Z date_updated: 2025-08-05T13:28:15Z day: '01' department: - _id: LaEr doi: 10.1007/s00440-023-01229-1 ec_funded: 1 external_id: arxiv: - '2210.12060' isi: - '001118972500001' intvolume: ' 188' isi: 1 language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2210.12060 month: '04' oa: 1 oa_version: Preprint page: 1131-1182 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Probability Theory and Related Fields publication_identifier: eissn: - 1432-2064 issn: - 0178-8051 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Mesoscopic central limit theorem for non-Hermitian random matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 188 year: '2024' ... --- DOAJ_listed: '1' OA_place: publisher OA_type: gold _id: '18762' abstract: - lang: eng text: Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance. acknowledgement: "I am very grateful to László Erdős for suggesting the topic and many valuable discussions during my work on the project. I would also like to thank the two anonymous referees for their careful reading of the manuscript and detailed feedback.\r\nPartially supported by ERC Advanced Grants “RMTBeyond” No. 101020331 and “LDRaM” No. 884584." article_number: '191' article_processing_charge: Yes article_type: original arxiv: 1 author: - first_name: Jana full_name: Reker, Jana id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9 last_name: Reker citation: ama: Reker J. Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. 2024;29. doi:10.1214/24-EJP1247 apa: Reker, J. (2024). Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/24-EJP1247 chicago: Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2024. https://doi.org/10.1214/24-EJP1247. ieee: J. Reker, “Multi-point functional central limit theorem for Wigner matrices,” Electronic Journal of Probability, vol. 29. Institute of Mathematical Statistics, 2024. ista: Reker J. 2024. Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. 29, 191. mla: Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” Electronic Journal of Probability, vol. 29, 191, Institute of Mathematical Statistics, 2024, doi:10.1214/24-EJP1247. short: J. Reker, Electronic Journal of Probability 29 (2024). corr_author: '1' date_created: 2025-01-05T23:01:58Z date_published: 2024-12-20T00:00:00Z date_updated: 2025-09-09T11:59:15Z day: '20' ddc: - '510' department: - _id: LaEr doi: 10.1214/24-EJP1247 ec_funded: 1 external_id: arxiv: - '2307.11028' isi: - '001381599200001' file: - access_level: open_access checksum: 67178feaa8630a332599d3037a3fe70e content_type: application/pdf creator: dernst date_created: 2025-01-08T08:44:03Z date_updated: 2025-01-08T08:44:03Z file_id: '18773' file_name: 2024_ElectrJournProbability_Reker.pdf file_size: 812428 relation: main_file success: 1 file_date_updated: 2025-01-08T08:44:03Z has_accepted_license: '1' intvolume: ' 29' isi: 1 language: - iso: eng month: '12' oa: 1 oa_version: Published Version project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Electronic Journal of Probability publication_identifier: eissn: - 1083-6489 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' related_material: record: - id: '17173' relation: earlier_version status: public scopus_import: '1' status: public title: Multi-point functional central limit theorem for 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 29 year: '2024' ... --- _id: '15025' abstract: - lang: eng text: We consider quadratic forms of deterministic matrices A evaluated at the random eigenvectors of a large N×N GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than √N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of A. Our result also naturally applies to the eigenvectors of any invariant ensemble. acknowledgement: The first author was supported by the ERC Advanced Grant “RMTBeyond” No. 101020331. The second author was supported by Fulbright Austria and the Austrian Marshall Plan Foundation. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Benjamin full_name: McKenna, Benjamin id: b0cc634c-d549-11ee-96c8-87338c7ad808 last_name: McKenna orcid: 0000-0003-2625-495X citation: ama: Erdös L, McKenna B. Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. 2024;34(1B):1623-1662. doi:10.1214/23-AAP2000 apa: Erdös, L., & McKenna, B. (2024). Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/23-AAP2000 chicago: Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms of GOE/GUE Eigenvectors.” Annals of Applied Probability. Institute of Mathematical Statistics, 2024. https://doi.org/10.1214/23-AAP2000. ieee: L. Erdös and B. McKenna, “Extremal statistics of quadratic forms of GOE/GUE eigenvectors,” Annals of Applied Probability, vol. 34, no. 1B. Institute of Mathematical Statistics, pp. 1623–1662, 2024. ista: Erdös L, McKenna B. 2024. Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. 34(1B), 1623–1662. mla: Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms of GOE/GUE Eigenvectors.” Annals of Applied Probability, vol. 34, no. 1B, Institute of Mathematical Statistics, 2024, pp. 1623–62, doi:10.1214/23-AAP2000. short: L. Erdös, B. McKenna, Annals of Applied Probability 34 (2024) 1623–1662. corr_author: '1' date_created: 2024-02-25T23:00:56Z date_published: 2024-02-01T00:00:00Z date_updated: 2025-09-04T12:08:11Z day: '01' department: - _id: LaEr doi: 10.1214/23-AAP2000 ec_funded: 1 external_id: arxiv: - '2208.12206' isi: - '001163006100021' intvolume: ' 34' isi: 1 issue: 1B language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2208.12206 month: '02' oa: 1 oa_version: Preprint page: 1623-1662 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Annals of Applied Probability publication_identifier: issn: - 1050-5164 publication_status: published publisher: Institute of Mathematical Statistics quality_controlled: '1' scopus_import: '1' status: public title: Extremal statistics of quadratic forms of GOE/GUE eigenvectors type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 34 year: '2024' ... --- OA_place: publisher OA_type: hybrid _id: '15378' abstract: - lang: eng text: We consider N×N non-Hermitian random matrices of the form X+A, where A is a general deterministic matrix and N−−√X consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, i.e. that the local density of eigenvalues is bounded by N1+o(1) and (ii) that the expected condition number of any bulk eigenvalue is bounded by N1+o(1); both results are optimal up to the factor No(1). The latter result complements the very recent matching lower bound obtained in [15] (arXiv:2301.03549) and improves the N-dependence of the upper bounds in [5,6,32] (arXiv:1906.11819, arXiv:2005.08930, arXiv:2005.08908). Our main ingredient, a near-optimal lower tail estimate for the small singular values of X+A−z, is of independent interest. acknowledgement: László Erdős is partially supported by ERC Advanced Grant “RMTBeyond” No. 101020331. Hong Chang Ji is supported by ERC Advanced Grant “RMTBeyond” No. 101020331. article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Hong Chang full_name: Ji, Hong Chang id: dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d last_name: Ji citation: ama: Erdös L, Ji HC. Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. 2024;77(9):3785-3840. doi:10.1002/cpa.22201 apa: Erdös, L., & Ji, H. C. (2024). Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.22201 chicago: Erdös, László, and Hong Chang Ji. “Wegner Estimate and Upper Bound on the Eigenvalue Condition Number of Non-Hermitian Random Matrices.” Communications on Pure and Applied Mathematics. Wiley, 2024. https://doi.org/10.1002/cpa.22201. ieee: L. Erdös and H. C. Ji, “Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices,” Communications on Pure and Applied Mathematics, vol. 77, no. 9. Wiley, pp. 3785–3840, 2024. ista: Erdös L, Ji HC. 2024. Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices. Communications on Pure and Applied Mathematics. 77(9), 3785–3840. mla: Erdös, László, and Hong Chang Ji. “Wegner Estimate and Upper Bound on the Eigenvalue Condition Number of Non-Hermitian Random Matrices.” Communications on Pure and Applied Mathematics, vol. 77, no. 9, Wiley, 2024, pp. 3785–840, doi:10.1002/cpa.22201. short: L. Erdös, H.C. Ji, Communications on Pure and Applied Mathematics 77 (2024) 3785–3840. corr_author: '1' date_created: 2024-05-12T22:01:02Z date_published: 2024-09-01T00:00:00Z date_updated: 2025-09-08T07:25:47Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1002/cpa.22201 ec_funded: 1 external_id: arxiv: - '2301.04981' isi: - '001217139900001' file: - access_level: open_access checksum: fbcc9cc7bf274f024e4f4afc9c208f96 content_type: application/pdf creator: dernst date_created: 2025-01-09T09:36:41Z date_updated: 2025-01-09T09:36:41Z file_id: '18803' file_name: 2024_CommPureApplMath_Erdoes.pdf file_size: 566963 relation: main_file success: 1 file_date_updated: 2025-01-09T09:36:41Z has_accepted_license: '1' intvolume: ' 77' isi: 1 issue: '9' language: - iso: eng license: https://creativecommons.org/licenses/by-nc-nd/4.0/ month: '09' oa: 1 oa_version: Published Version page: 3785-3840 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Communications on Pure and Applied Mathematics publication_identifier: eissn: - 1097-0312 issn: - 0010-3640 publication_status: published publisher: Wiley quality_controlled: '1' scopus_import: '1' status: public title: Wegner estimate and upper bound on the eigenvalue condition number of non-Hermitian random matrices tmp: image: /images/cc_by_nc_nd.png legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) short: CC BY-NC-ND (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 77 year: '2024' ... --- _id: '17281' abstract: - lang: eng text: We extend the free convolution of Brown measures of R-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions. acknowledgement: This work was supported by the National Science Foundation [Grant No. DMS-2143142 to S.O.]; and the European Research Council [Grant No. 101020331].The third author acknowledges the support of the University of Colorado Boulder, where a portion of this work was completed. The authors thank Martin Auer, Vadim Gorin, Brian Hall, and Noah Williams for comments, corrections, and references. The authors also wish to thank the anonymous referees for useful feedback and corrections. article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Andrew J full_name: Campbell, Andrew J id: 582b06a9-1f1c-11ee-b076-82ffce00dde4 last_name: Campbell - first_name: Sean full_name: O'Rourke, Sean last_name: O'Rourke - first_name: David T full_name: Renfrew, David T id: 4845BF6A-F248-11E8-B48F-1D18A9856A87 last_name: Renfrew orcid: 0000-0003-3493-121X citation: ama: Campbell AJ, O’Rourke S, Renfrew DT. The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation. International Mathematics Research Notices. 2024;2024(13):10189-10218. doi:10.1093/imrn/rnae062 apa: Campbell, A. J., O’Rourke, S., & Renfrew, D. T. (2024). The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnae062 chicago: Campbell, Andrew J, Sean O’Rourke, and David T Renfrew. “The Fractional Free Convolution of R-Diagonal Elements and Random Polynomials under Repeated Differentiation.” International Mathematics Research Notices. Oxford University Press, 2024. https://doi.org/10.1093/imrn/rnae062. ieee: A. J. Campbell, S. O’Rourke, and D. T. Renfrew, “The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation,” International Mathematics Research Notices, vol. 2024, no. 13. Oxford University Press, pp. 10189–10218, 2024. ista: Campbell AJ, O’Rourke S, Renfrew DT. 2024. The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation. International Mathematics Research Notices. 2024(13), 10189–10218. mla: Campbell, Andrew J., et al. “The Fractional Free Convolution of R-Diagonal Elements and Random Polynomials under Repeated Differentiation.” International Mathematics Research Notices, vol. 2024, no. 13, Oxford University Press, 2024, pp. 10189–218, doi:10.1093/imrn/rnae062. short: A.J. Campbell, S. O’Rourke, D.T. Renfrew, International Mathematics Research Notices 2024 (2024) 10189–10218. corr_author: '1' date_created: 2024-07-21T22:01:01Z date_published: 2024-07-01T00:00:00Z date_updated: 2025-09-08T08:16:32Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1093/imrn/rnae062 external_id: isi: - '001198019500001' file: - access_level: open_access checksum: f36a7dbf53f23d5833db711052e69b49 content_type: application/pdf creator: dernst date_created: 2024-07-22T06:40:19Z date_updated: 2024-07-22T06:40:19Z file_id: '17288' file_name: 2024_IMRN_Campbell.pdf file_size: 1233508 relation: main_file success: 1 file_date_updated: 2024-07-22T06:40:19Z has_accepted_license: '1' intvolume: ' 2024' isi: 1 issue: '13' language: - iso: eng month: '07' oa: 1 oa_version: Published Version page: 10189-10218 publication: International Mathematics Research Notices publication_identifier: eissn: - 1687-0247 issn: - 1073-7928 publication_status: published publisher: Oxford University Press quality_controlled: '1' scopus_import: '1' status: public title: The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 2024 year: '2024' ... --- _id: '17375' abstract: - lang: eng text: We consider the spectral radius of a large random matrix X with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of X in Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of X − z for different complex shift parameters z using the Dyson Brownian Motion. acknowledgement: L.E. and Y.X. were supported by the ERC Advanced Grant “RMTBeyond” Grant No. 101020331. article_number: '063302' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Giorgio full_name: Cipolloni, Giorgio id: 42198EFA-F248-11E8-B48F-1D18A9856A87 last_name: Cipolloni orcid: 0000-0002-4901-7992 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Yuanyuan full_name: Xu, Yuanyuan id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3 last_name: Xu orcid: 0000-0003-1559-1205 citation: ama: Cipolloni G, Erdös L, Xu Y. Precise asymptotics for the spectral radius of a large random matrix. Journal of Mathematical Physics. 2024;65(6). doi:10.1063/5.0209705 apa: Cipolloni, G., Erdös, L., & Xu, Y. (2024). Precise asymptotics for the spectral radius of a large random matrix. Journal of Mathematical Physics. AIP Publishing. https://doi.org/10.1063/5.0209705 chicago: Cipolloni, Giorgio, László Erdös, and Yuanyuan Xu. “Precise Asymptotics for the Spectral Radius of a Large Random Matrix.” Journal of Mathematical Physics. AIP Publishing, 2024. https://doi.org/10.1063/5.0209705. ieee: G. Cipolloni, L. Erdös, and Y. Xu, “Precise asymptotics for the spectral radius of a large random matrix,” Journal of Mathematical Physics, vol. 65, no. 6. AIP Publishing, 2024. ista: Cipolloni G, Erdös L, Xu Y. 2024. Precise asymptotics for the spectral radius of a large random matrix. Journal of Mathematical Physics. 65(6), 063302. mla: Cipolloni, Giorgio, et al. “Precise Asymptotics for the Spectral Radius of a Large Random Matrix.” Journal of Mathematical Physics, vol. 65, no. 6, 063302, AIP Publishing, 2024, doi:10.1063/5.0209705. short: G. Cipolloni, L. Erdös, Y. Xu, Journal of Mathematical Physics 65 (2024). corr_author: '1' date_created: 2024-08-04T22:01:22Z date_published: 2024-06-01T00:00:00Z date_updated: 2025-09-08T08:44:57Z day: '01' department: - _id: LaEr doi: 10.1063/5.0209705 ec_funded: 1 external_id: arxiv: - '2210.15643' isi: - '001252240700002' intvolume: ' 65' isi: 1 issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2210.15643 month: '06' oa: 1 oa_version: Preprint project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Journal of Mathematical Physics publication_identifier: issn: - 0022-2488 publication_status: published publisher: AIP Publishing quality_controlled: '1' scopus_import: '1' status: public title: Precise asymptotics for the spectral radius of a large random matrix type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 65 year: '2024' ... --- OA_place: publisher OA_type: hybrid _id: '18554' abstract: - lang: eng text: We prove the Eigenstate Thermalization Hypothesis for general Wigner-type matrices in the bulk of the self-consistent spectrum, with optimal control on the fluctuations for obs ervables of arbitrary rank. As the main technical ingredient, we prove rank-uniform optimal local laws for one and two resolvents of a Wigner-type matrix with regular observables. Our results hold under very general conditions on the variance profile, even allowing many vanishing entries, demonstrating that Eigenstate Thermalization occurs robustly across a diverse class of random matrix ensembles, for which the underlying quantum system has a non-trivial spatial structure. acknowledgement: Open access funding provided by Institute of Science and Technology (IST Austria). article_number: '282' article_processing_charge: Yes (via OA deal) article_type: original arxiv: 1 author: - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Volodymyr full_name: Riabov, Volodymyr id: 1949f904-edfb-11eb-afb5-e2dfddabb93b last_name: Riabov citation: ama: Erdös L, Riabov V. Eigenstate Thermalization Hypothesis for Wigner-type matrices. Communications in Mathematical Physics. 2024;405(12). doi:10.1007/s00220-024-05143-y apa: Erdös, L., & Riabov, V. (2024). Eigenstate Thermalization Hypothesis for Wigner-type matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-024-05143-y chicago: Erdös, László, and Volodymyr Riabov. “Eigenstate Thermalization Hypothesis for Wigner-Type Matrices.” Communications in Mathematical Physics. Springer Nature, 2024. https://doi.org/10.1007/s00220-024-05143-y. ieee: L. Erdös and V. Riabov, “Eigenstate Thermalization Hypothesis for Wigner-type matrices,” Communications in Mathematical Physics, vol. 405, no. 12. Springer Nature, 2024. ista: Erdös L, Riabov V. 2024. Eigenstate Thermalization Hypothesis for Wigner-type matrices. Communications in Mathematical Physics. 405(12), 282. mla: Erdös, László, and Volodymyr Riabov. “Eigenstate Thermalization Hypothesis for Wigner-Type Matrices.” Communications in Mathematical Physics, vol. 405, no. 12, 282, Springer Nature, 2024, doi:10.1007/s00220-024-05143-y. short: L. Erdös, V. Riabov, Communications in Mathematical Physics 405 (2024). corr_author: '1' date_created: 2024-11-17T23:01:46Z date_published: 2024-12-01T00:00:00Z date_updated: 2026-04-07T12:32:19Z day: '01' ddc: - '510' department: - _id: LaEr doi: 10.1007/s00220-024-05143-y external_id: arxiv: - '2403.10359' isi: - '001348943900004' pmid: - '39526190' file: - access_level: open_access checksum: c9ae0ea195bd39b8b3a630d492fb00dc content_type: application/pdf creator: dernst date_created: 2024-11-18T08:15:07Z date_updated: 2024-11-18T08:15:07Z file_id: '18562' file_name: 2024_CommMathPhysics_Erdoes.pdf file_size: 1426046 relation: main_file success: 1 file_date_updated: 2024-11-18T08:15:07Z has_accepted_license: '1' intvolume: ' 405' isi: 1 issue: '12' language: - iso: eng month: '12' oa: 1 oa_version: Published Version pmid: 1 publication: Communications in Mathematical Physics publication_identifier: eissn: - 1432-0916 issn: - 0010-3616 publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: record: - id: '20575' relation: dissertation_contains status: public scopus_import: '1' status: public title: Eigenstate Thermalization Hypothesis for 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 405 year: '2024' ... --- OA_place: repository OA_type: green _id: '18656' abstract: - lang: eng text: "We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables \r\n and \r\n in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [10]) or the Hilbert-Schmidt norm (cf. [11])." acknowledgement: LE and JH were supported by the ERC Advanced Grant łRMTBeyondž No. 101020331 article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Giorgio full_name: Cipolloni, Giorgio id: 42198EFA-F248-11E8-B48F-1D18A9856A87 last_name: Cipolloni orcid: 0000-0002-4901-7992 - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Sven Joscha full_name: Henheik, Sven Joscha id: 31d731d7-d235-11ea-ad11-b50331c8d7fb last_name: Henheik orcid: 0000-0003-1106-327X citation: ama: Cipolloni G, Erdös L, Henheik SJ. Out-of-time-ordered correlators for Wigner matrices. Advances in Theoretical and Mathematical Physics. 2024;28(6):2025-2083. doi:10.4310/ATMP.241031013250 apa: Cipolloni, G., Erdös, L., & Henheik, S. J. (2024). Out-of-time-ordered correlators for Wigner matrices. Advances in Theoretical and Mathematical Physics. International Press. https://doi.org/10.4310/ATMP.241031013250 chicago: Cipolloni, Giorgio, László Erdös, and Sven Joscha Henheik. “Out-of-Time-Ordered Correlators for Wigner Matrices.” Advances in Theoretical and Mathematical Physics. International Press, 2024. https://doi.org/10.4310/ATMP.241031013250. ieee: G. Cipolloni, L. Erdös, and S. J. Henheik, “Out-of-time-ordered correlators for Wigner matrices,” Advances in Theoretical and Mathematical Physics, vol. 28, no. 6. International Press, pp. 2025–2083, 2024. ista: Cipolloni G, Erdös L, Henheik SJ. 2024. Out-of-time-ordered correlators for Wigner matrices. Advances in Theoretical and Mathematical Physics. 28(6), 2025–2083. mla: Cipolloni, Giorgio, et al. “Out-of-Time-Ordered Correlators for Wigner Matrices.” Advances in Theoretical and Mathematical Physics, vol. 28, no. 6, International Press, 2024, pp. 2025–83, doi:10.4310/ATMP.241031013250. short: G. Cipolloni, L. Erdös, S.J. Henheik, Advances in Theoretical and Mathematical Physics 28 (2024) 2025–2083. corr_author: '1' date_created: 2024-12-15T23:01:51Z date_published: 2024-10-30T00:00:00Z date_updated: 2026-04-07T12:37:10Z day: '30' department: - _id: LaEr doi: 10.4310/ATMP.241031013250 ec_funded: 1 external_id: arxiv: - '2402.17609' intvolume: ' 28' issue: '6' language: - iso: eng main_file_link: - open_access: '1' url: https://doi.org/10.48550/arXiv.2402.17609 month: '10' oa: 1 oa_version: Preprint page: 2025-2083 project: - _id: 62796744-2b32-11ec-9570-940b20777f1d call_identifier: H2020 grant_number: '101020331' name: Random matrices beyond Wigner-Dyson-Mehta publication: Advances in Theoretical and Mathematical Physics publication_identifier: eissn: - 1095-0753 issn: - 1095-0761 publication_status: published publisher: International Press quality_controlled: '1' related_material: record: - id: '19540' relation: dissertation_contains status: public scopus_import: '1' status: public title: Out-of-time-ordered correlators for Wigner matrices type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 28 year: '2024' ...