论文标题
麦克唐纳（MacDonald
Noncolliding Macdonald Walks with an Absorbing Wall
论文作者
论文摘要
分支规则是麦克唐纳对称多项式的最基本特性之一。它表示麦克唐纳多项式作为麦克唐纳多项式的非负线性组合，变量较少。当变量数量输入无穷大时，在主专业中限制了分支规则的限制，我们获得了一个$ M $非收缩粒子的Markov链，其颗粒为负漂移和零的吸收壁。该链取决于麦克唐纳参数$（q，t）$，并且可能被视为戴森·布朗尼运动的离散变形。马尔可夫链的轨迹等同于具有任意级联前壁的平面分区的某些吉布斯集合。在千斤顶限制中，$ t = q^{β/2} \至1 $，吸收墙消失了，麦克唐纳非倾斜步行将$β$ - noncollising随机散步变成了Huang [Int]研究的随机步行。数学。 res。不是。 2021（2021），5898-5942，Arxiv：1708.07115]。采用$ q = 0 $（Hall-Littlewood变性），然后进一步发送$ t \至1 $，我们在$ \ mathbb {z} _ {\ ge 0} $上获得了连续的时间粒子系统，并以不均匀的跳跃速率和零吸收壁。
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 $t=q^{β/2}\to 1$ 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 $t\to 1$, we obtain a continuous time particle system on $\mathbb{Z}_{\ge 0}$ with inhomogeneous jump rates and absorbing wall at zero.
