Limits and fluctuations of p-adic random matrix products
We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over Qp are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex...
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| Format: | Article |
| Language: | English |
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Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/132769 |
| _version_ | 1826203977815949312 |
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| author | Van Peski, Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Van Peski, Roger |
| author_sort | Van Peski, Roger |
| collection | MIT |
| description | We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over Qp are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of GLN(Zp) form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of GLN(Zp) corners. |
| first_indexed | 2024-09-23T12:46:54Z |
| format | Article |
| id | mit-1721.1/132769 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:46:54Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1327692024-06-03T17:28:25Z Limits and fluctuations of p-adic random matrix products Van Peski, Roger Massachusetts Institute of Technology. Department of Mathematics We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over Qp are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of GLN(Zp) form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of GLN(Zp) corners. 2021-10-07T14:28:54Z 2021-10-07T14:28:54Z 2021-10 2021-10-07T03:33:31Z Article http://purl.org/eprint/type/JournalArticle 1022-1824 1420-9020 https://hdl.handle.net/1721.1/132769 Van Peski, R. Limits and fluctuations of p-adic random matrix products. Sel. Math. New Ser. 27, 98 (2021) en 10.1007/s00029-021-00709-3 Selecta Mathematica Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Van Peski, Roger Limits and fluctuations of p-adic random matrix products |
| title | Limits and fluctuations of p-adic random matrix products |
| title_full | Limits and fluctuations of p-adic random matrix products |
| title_fullStr | Limits and fluctuations of p-adic random matrix products |
| title_full_unstemmed | Limits and fluctuations of p-adic random matrix products |
| title_short | Limits and fluctuations of p-adic random matrix products |
| title_sort | limits and fluctuations of p adic random matrix products |
| url | https://hdl.handle.net/1721.1/132769 |
| work_keys_str_mv | AT vanpeskiroger limitsandfluctuationsofpadicrandommatrixproducts |