Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 079, 21 pages      arXiv:2204.09206      https://doi.org/10.3842/SIGMA.2022.079

Noncolliding Macdonald Walks with an Absorbing Wall

Leonid Petrov
University of Virginia, Charlottesville, VA, USA

Received June 07, 2022, in final form October 16, 2022; Published online October 20, 2022

Abstract
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 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 $m$ noncolliding particles with negative drift and an absorbing wall at zero. The chain depends on the Macdonald parameters $(q,t)$ 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 $t=q^{\beta/2}\to1$ the absorbing wall disappears, and the Macdonald noncolliding walks turn into the $\beta$-noncolliding random walks studied by Huang [Int. Math. Res. Not. 2021 (2021), 5898-5942, arXiv:1708.07115]. Taking $q=0$ (Hall-Littlewood degeneration) and further sending $t\to 1$, we obtain a continuous time particle system on $\mathbb{Z}_{\ge 0}$ with inhomogeneous jump rates and absorbing wall at zero.

Key words: Macdonald polynomials; branching rule; noncolliding random walks; lozenge tilings.

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