Noncolliding Macdonald Walks with an Absorbing Wall
The branching rule is one of the most fundamental properties of the Macdonald symmetric polynomials. It expresses a Macdonald polynomial as a nonnegative linear combination of Macdonald polynomials with a smaller number of variables. Taking a limit of the branching rule under the principal specializ...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211825 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Noncolliding Macdonald Walks with an Absorbing Wall. Leonid Petrov. SIGMA 18 (2022), 079, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862628524579356672 |
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| author | Petrov, Leonid |
| author_facet | Petrov, Leonid |
| citation_txt | Noncolliding Macdonald Walks with an Absorbing Wall. Leonid Petrov. SIGMA 18 (2022), 079, 21 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The branching rule is one of the most fundamental properties of the Macdonald symmetric polynomials. It expresses a Macdonald polynomial as a nonnegative linear combination of Macdonald polynomials with a smaller number of variables. Taking a limit of the branching rule under the principal specialization when the number of variables goes to infinity, we obtain a Markov chain of noncolliding particles with negative drift and an absorbing wall at zero. The chain depends on the Macdonald parameters (, ) and may be viewed as a discrete deformation of the Dyson Brownian motion. The trajectory of the Markov chain is equivalent to a certain Gibbs ensemble of plane partitions with an arbitrary cascade front wall. In the Jack limit = ᵝᐟ² → 1, the absorbing wall disappears, and the Macdonald noncolliding walks turn into the β-noncolliding random walks studied by Huang [Int. Math. Res. Not. 2021 (2021), 5898-5942, arXiv:1708.07115]. Taking = 0 (Hall-Littlewood degeneration) and further sending → 1, we obtain a continuous time particle system on ℤ≥₀ with inhomogeneous jump rates and absorbing wall at zero.
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| first_indexed | 2026-03-14T17:18:09Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211825 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T17:18:09Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
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| spelling | Petrov, Leonid 2026-01-12T10:21:21Z 2022 Noncolliding Macdonald Walks with an Absorbing Wall. Leonid Petrov. SIGMA 18 (2022), 079, 21 pages 1815-0659 2020 Mathematics Subject Classification: 06C05; 05E05; 05A30 arXiv:2204.09206 https://nasplib.isofts.kiev.ua/handle/123456789/211825 https://doi.org/10.3842/SIGMA.2022.079 The branching rule is one of the most fundamental properties of the Macdonald symmetric polynomials. It expresses a Macdonald polynomial as a nonnegative linear combination of Macdonald polynomials with a smaller number of variables. Taking a limit of the branching rule under the principal specialization when the number of variables goes to infinity, we obtain a Markov chain of noncolliding particles with negative drift and an absorbing wall at zero. The chain depends on the Macdonald parameters (, ) and may be viewed as a discrete deformation of the Dyson Brownian motion. The trajectory of the Markov chain is equivalent to a certain Gibbs ensemble of plane partitions with an arbitrary cascade front wall. In the Jack limit = ᵝᐟ² → 1, the absorbing wall disappears, and the Macdonald noncolliding walks turn into the β-noncolliding random walks studied by Huang [Int. Math. Res. Not. 2021 (2021), 5898-5942, arXiv:1708.07115]. Taking = 0 (Hall-Littlewood degeneration) and further sending → 1, we obtain a continuous time particle system on ℤ≥₀ with inhomogeneous jump rates and absorbing wall at zero. I am grateful to Alexei Borodin, Grigori Olshanski, and Mikhail Tikhonov for fruitful discussions, and to the anonymous referees for helpful remarks. The work was partially supported by the NSFgrant DMS-1664617, and the Simons Collaboration Grant for Mathematicians 709055. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930, while LP participated in the program “Universality and Integrability in random matrix theory and Interacting Particle Systems” hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Noncolliding Macdonald Walks with an Absorbing Wall Article published earlier |
| spellingShingle | Noncolliding Macdonald Walks with an Absorbing Wall Petrov, Leonid |
| title | Noncolliding Macdonald Walks with an Absorbing Wall |
| title_full | Noncolliding Macdonald Walks with an Absorbing Wall |
| title_fullStr | Noncolliding Macdonald Walks with an Absorbing Wall |
| title_full_unstemmed | Noncolliding Macdonald Walks with an Absorbing Wall |
| title_short | Noncolliding Macdonald Walks with an Absorbing Wall |
| title_sort | noncolliding macdonald walks with an absorbing wall |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211825 |
| work_keys_str_mv | AT petrovleonid noncollidingmacdonaldwalkswithanabsorbingwall |