Noncolliding Macdonald Walks with an Absorbing Wall

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 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 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 = 𝑞ᵝᐟ² → 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 𝑞 = 0 (Hall-Littlewood degeneration) and further sending 𝘵 → 1, we obtain a continuous time particle system on ℤ≥₀ with inhomogeneous jump rates and absorbing wall at zero.