Noncolliding Macdonald Walks with an Absorbing Wall

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• ISSN: 1815-0659
• Main Author: Petrov, Leonid
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211825
• 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

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.