Spectral theory for the q-Boson particle system
We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enab...
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| Format: | Article |
| Language: | en_US |
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Cambridge University Press
2016
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| Online Access: | http://hdl.handle.net/1721.1/104846 https://orcid.org/0000-0002-2913-5238 |
| _version_ | 1826209016689197056 |
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| author | Borodin, Alexei Corwin, Ivan Petrov, Leonid Sasamoto, Tomohiro |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei Corwin, Ivan Petrov, Leonid Sasamoto, Tomohiro |
| author_sort | Borodin, Alexei |
| collection | MIT |
| description | We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP (q-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation. |
| first_indexed | 2024-09-23T14:16:21Z |
| format | Article |
| id | mit-1721.1/104846 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:16:21Z |
| publishDate | 2016 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | mit-1721.1/1048462022-10-01T20:16:08Z Spectral theory for the q-Boson particle system Borodin, Alexei Corwin, Ivan Petrov, Leonid Sasamoto, Tomohiro Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei Corwin, Ivan We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP (q-deformed totally asymmetric simple exclusion process), this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell–Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation/Kardar–Parisi–Zhang equation. 2016-10-19T14:30:32Z 2016-10-19T14:30:32Z 2014-09 2013-10 Article http://purl.org/eprint/type/JournalArticle 0010-437X 1570-5846 http://hdl.handle.net/1721.1/104846 Borodin, Alexei, Ivan Corwin, Leonid Petrov, and Tomohiro Sasamoto. “Spectral Theory for the q-Boson Particle System.” Compositio Math. 151, no. 01 (September 17, 2014): 1–67. https://orcid.org/0000-0002-2913-5238 en_US http://dx.doi.org/10.1112/s0010437x14007532 Compositio Mathematica Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Cambridge University Press arXiv |
| spellingShingle | Borodin, Alexei Corwin, Ivan Petrov, Leonid Sasamoto, Tomohiro Spectral theory for the q-Boson particle system |
| title | Spectral theory for the q-Boson particle system |
| title_full | Spectral theory for the q-Boson particle system |
| title_fullStr | Spectral theory for the q-Boson particle system |
| title_full_unstemmed | Spectral theory for the q-Boson particle system |
| title_short | Spectral theory for the q-Boson particle system |
| title_sort | spectral theory for the q boson particle system |
| url | http://hdl.handle.net/1721.1/104846 https://orcid.org/0000-0002-2913-5238 |
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