非碰撞的麦克唐纳与可吸收的墙壁一起行走

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS
L. Petrov
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引用次数: 1

摘要

分支规则是麦克唐纳对称多项式最基本的性质之一。它将麦克唐纳多项式表示为变量数量较少的麦克唐纳多项式的非负线性组合。当变量数量无穷大时,在主专门化下取分支规则的一个极限,我们得到了一个具有负漂移和零吸收壁的$m$非碰撞粒子的马尔可夫链。该链取决于麦克唐纳参数$(q,t)$,并且可以被视为戴森布朗运动的离散变形。马尔可夫链的轨迹等价于具有任意级联前壁的平面分区的某个吉布斯系综。在Jack极限$t=q^{\beta/2}\为1$时,吸收壁消失,Macdonald非碰撞行走转变为Huang研究的$\beta$非碰撞随机行走[Int.Math.Res.Not.2021(2021),5898-5942,arXiv:170807115]。取$q=0$(Hall-Littlewood退化)并进一步将$t\发送到1$,我们在$\mathbb上获得了一个连续时间粒子系统{Z}_{\ge 0}$,具有不均匀的跳跃率和零处的吸收壁。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Noncolliding Macdonald Walks with an Absorbing Wall
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.
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
期刊介绍: Scope Geometrical methods in mathematical physics Lie theory and differential equations Classical and quantum integrable systems Algebraic methods in dynamical systems and chaos Exactly and quasi-exactly solvable models Lie groups and algebras, representation theory Orthogonal polynomials and special functions Integrable probability and stochastic processes Quantum algebras, quantum groups and their representations Symplectic, Poisson and noncommutative geometry Algebraic geometry and its applications Quantum field theories and string/gauge theories Statistical physics and condensed matter physics Quantum gravity and cosmology.
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