Sums of GUE matrices and concentration of hives from correlation decay of eigengaps
Associated to two given sequences of eigenvalues λ 1 ≥ ⋯ ≥ λ n and μ 1 ≥ ⋯ ≥ μ n is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis o...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2024
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| Online Access: | https://hdl.handle.net/1721.1/157405 |
| _version_ | 1824458248831369216 |
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| author | Narayanan, Hariharan Sheffield, Scott Tao, Terence |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Narayanan, Hariharan Sheffield, Scott Tao, Terence |
| author_sort | Narayanan, Hariharan |
| collection | MIT |
| description | Associated to two given sequences of eigenvalues λ 1 ≥ ⋯ ≥ λ n and μ 1 ≥ ⋯ ≥ μ n is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as n → ∞ . Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdős–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. |
| first_indexed | 2025-02-19T04:22:53Z |
| format | Article |
| id | mit-1721.1/157405 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2025-02-19T04:22:53Z |
| publishDate | 2024 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1574052025-01-07T04:38:33Z Sums of GUE matrices and concentration of hives from correlation decay of eigengaps Narayanan, Hariharan Sheffield, Scott Tao, Terence Massachusetts Institute of Technology. Department of Mathematics Associated to two given sequences of eigenvalues λ 1 ≥ ⋯ ≥ λ n and μ 1 ≥ ⋯ ≥ μ n is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as n → ∞ . Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdős–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. 2024-10-23T15:40:41Z 2024-10-23T15:40:41Z 2023-12-28 2024-10-19T03:40:02Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/157405 Narayanan, H., Sheffield, S. & Tao, T. Sums of GUE matrices and concentration of hives from correlation decay of eigengaps. Probab. Theory Relat. Fields 190, 1121–1165 (2024). en https://doi.org/10.1007/s00440-023-01250-4 Probability Theory and Related Fields Creative Commons Attribution-Noncommercial-ShareAlike http://creativecommons.org/licenses/by-nc-sa/4.0/ The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Narayanan, Hariharan Sheffield, Scott Tao, Terence Sums of GUE matrices and concentration of hives from correlation decay of eigengaps |
| title | Sums of GUE matrices and concentration of hives from correlation decay of eigengaps |
| title_full | Sums of GUE matrices and concentration of hives from correlation decay of eigengaps |
| title_fullStr | Sums of GUE matrices and concentration of hives from correlation decay of eigengaps |
| title_full_unstemmed | Sums of GUE matrices and concentration of hives from correlation decay of eigengaps |
| title_short | Sums of GUE matrices and concentration of hives from correlation decay of eigengaps |
| title_sort | sums of gue matrices and concentration of hives from correlation decay of eigengaps |
| url | https://hdl.handle.net/1721.1/157405 |
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