Noncolliding Macdonald walks with an absorbing wall


Leonid Petrov
SIGMA 18 (2022), 079, 21 pagesarXiv:2204.09206 [math.PR]


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 $q=t^{\beta/2}\to1$ the absorbing wall disappears, and the Macdonald noncolliding walks turn into the $\beta$-noncolliding random walks studied by Huang [arXiv:1708.07115]. Taking $q=0$ (Hall-Littlewood degeneration) and further sending $t\to 1$, we obtain a continuous time particle system on $\mathbb{Z}_{\ge0}$ with inhomogeneous jump rates and absorbing wall at zero.

Noncolliding walks, lozenge tilings, and plane partitions
Noncolliding walks, lozenge tilings, and plane partitions