Stochastic Airy semigroup through tridiagonal matrices
We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy β process, which describes the largest eigenvalues in th...
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| Format: | Article |
| Language: | English |
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Institute of Mathematical Statistics
2020
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| Online Access: | https://hdl.handle.net/1721.1/123482 |
| _version_ | 1811092038453559296 |
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| author | Gorin, Vadim Shkolnikov, Mykhaylo |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Gorin, Vadim Shkolnikov, Mykhaylo |
| author_sort | Gorin, Vadim |
| collection | MIT |
| description | We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy β process, which describes the largest eigenvalues in the β ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable. Keywords: Airy point process; Brownian bridge; Brownian excursion; Dumitriu–Edelman model; Feynman–Kac formula; Gaussian beta ensemble; intersection local time; moment method; path transformation; quantile transform; random matrix soft edge; random walk bridge; stochastic Airy operator; strong invariance principle; trace formula; Vervaat transform |
| first_indexed | 2024-09-23T15:11:56Z |
| format | Article |
| id | mit-1721.1/123482 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:11:56Z |
| publishDate | 2020 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1234822022-10-02T01:18:18Z Stochastic Airy semigroup through tridiagonal matrices Gorin, Vadim Shkolnikov, Mykhaylo Massachusetts Institute of Technology. Department of Mathematics Statistics, Probability and Uncertainty Statistics and Probability We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy β process, which describes the largest eigenvalues in the β ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable. Keywords: Airy point process; Brownian bridge; Brownian excursion; Dumitriu–Edelman model; Feynman–Kac formula; Gaussian beta ensemble; intersection local time; moment method; path transformation; quantile transform; random matrix soft edge; random walk bridge; stochastic Airy operator; strong invariance principle; trace formula; Vervaat transform National Science Foundation (U.S.) (Grant DMS-1407562) National Science Foundation (U.S.) (Grant DMS-1664619) 2020-01-20T22:00:46Z 2020-01-20T22:00:46Z 2018-06 2016-06 2019-11-13T15:21:07Z Article http://purl.org/eprint/type/JournalArticle 0091-1798 2168-894X https://hdl.handle.net/1721.1/123482 Gorin, Vadim and Shkolnikov, Mykhaylo. "Stochastic Airy semigroup through tridiagonal matrices." Annals of Probability, 46, no.4, (2018): 2287--2344 © Institute of Mathematical Statistics, 2018. en https://doi.org/10.1214/17-aop1229 The Annals of Probability Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Statistics, Probability and Uncertainty Statistics and Probability Gorin, Vadim Shkolnikov, Mykhaylo Stochastic Airy semigroup through tridiagonal matrices |
| title | Stochastic Airy semigroup through tridiagonal matrices |
| title_full | Stochastic Airy semigroup through tridiagonal matrices |
| title_fullStr | Stochastic Airy semigroup through tridiagonal matrices |
| title_full_unstemmed | Stochastic Airy semigroup through tridiagonal matrices |
| title_short | Stochastic Airy semigroup through tridiagonal matrices |
| title_sort | stochastic airy semigroup through tridiagonal matrices |
| topic | Statistics, Probability and Uncertainty Statistics and Probability |
| url | https://hdl.handle.net/1721.1/123482 |
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