Noncolliding Macdonald Walks with an Absorbing Wall

TL;DR

This paper introduces a new Markov chain derived from Macdonald polynomials that models noncolliding particles with an absorbing wall, connecting algebraic structures to stochastic processes and their limits.

Contribution

It develops a novel Markov chain from Macdonald polynomials, linking algebraic combinatorics with noncolliding particle systems and their continuous limits.

Findings

The chain depends on Macdonald parameters (q,t) and generalizes Dyson Brownian motion.

In the Jack limit, the model reduces to beta-noncolliding random walks.

Degenerations yield continuous-time particle systems with inhomogeneous jump rates.

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 $t=q^{\beta/2}\to 1$ 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.