论文标题
$ q $ -pushtasep的下尾
The lower tail of $q$-pushTASEP
论文作者
论文摘要
我们研究$ q $ -pushtasep，这是一种离散的时间交互粒子系统，其分布与$ q $ - 惠泰克度量有关。我们证明，从步骤初始条件启动时，在$ n $下的尾部在波动量表上的下降范围$ x_n（n）$。我们的论点依赖于从$ q $ whittaker度量的地图到定期上次通道渗透模型（LPP），并在无限条纹中具有几何量，该地图最近在[Arxiv：2106.11922]中建立。通过路径路由参数，我们将标准LPP的独立通道时间的无限总和在$ n \ times n $方面的无限总和与几何量衰减的几何衰减。为了证明我们的尾巴结合结果，我们将降低与浓度不等式相结合，以及至关重要的新技术结果 - 在$ n \ times n $ last通道上的较低尾巴范围均匀地均匀地比\ mathbb n $中的所有$ n \ y Mathbb n $以及所有的几何参数（0,1）$。该技术结果使用Widom的技巧[ARXIV：MATH/0108008]，并适应了为GUE [LED05A]引入的Ledoux的想法，以减少均匀的下尾巴，以非常高的渐近矩与均匀的渐近矩束缚，最多订购了$ n $，最多是Meixner Ensemble。我们通过首先从Ledoux [LED05B]的显式组合公式中获得Meixner集合的阶乘时刻的尖锐均匀估计来实现这一目标，并通过进一步的仔细分析和精致的消除来将其转化为多项式界限。
We study $q$-pushTASEP, a discrete time interacting particle system whose distribution is related to the $q$-Whittaker measure. We prove a uniform in $N$ lower tail bound on the fluctuation scale for the location $x_N(N)$ of the right-most particle at time $N$ when started from step initial condition. Our argument relies on a map from the $q$-Whittaker measure to a model of periodic last passage percolation (LPP) with geometric weights in an infinite strip that was recently established in [arXiv:2106.11922]. By a path routing argument we bound the passage time in the periodic environment in terms of an infinite sum of independent passage times for standard LPP on $N\times N$ squares with geometric weights whose parameters decay geometrically. To prove our tail bound result we combine this reduction with a concentration inequality, and a crucial new technical result -- lower tail bounds on $N\times N$ last passage times uniformly over all $N \in \mathbb N$ and all the geometric parameters in $(0,1)$. This technical result uses Widom's trick [arXiv:math/0108008] and an adaptation of an idea of Ledoux introduced for the GUE [Led05a] to reduce the uniform lower tail bound to uniform asymptotics for very high moments, up to order $N$, of the Meixner ensemble. This we accomplish by first obtaining sharp uniform estimates for factorial moments of the Meixner ensemble from an explicit combinatorial formula of Ledoux [Led05b], and translating them to polynomial bounds via a further careful analysis and delicate cancellation.
