In a series of papers [23], [24], [25] by Bufetov and Gorin, Schur generating functions as the Fourier transforms on the unitary group $U(N)$, are introduced to study the asymptotic behaviors of random N-particle systems. We introduce and study the Jack generating functions of random N-particle systems. In special cases, this can be viewed as the Fourier transforms on the Gelfand pairs $(G{L}_N(\mathbb{R}),O(N))$, $(G{L}_N(\mathbb{C}),U(N))$ and $(G{L}_N(\mathbb{H}),Sp(N))$. Our main results state that the law of large numbers and the central limit theorems for such particle systems, is equivalent to certain conditions on the germ at unity of their Jack generating functions. Our results provide a new toolbox for the analysis of the global behavior of stochastic discrete particle systems via their Jack generating functions, and the limiting behaviors of Jack symmetric functions via stochastic discrete dynamics. Some applications of our main results include:

1.

Using the law of large numbers and the central limit theorems for β-nonintersecting Poisson random walks in [33] as input, our theorem gives the asymptotics of Jack characters at unity.

2.

We prove law of large numbers and central limit theorems for the Littlewood-Richardson coefficients of zonal polynomials.

3.

We show that the fluctuations of the height functions of a general family of nonintersecting random walks are asymptotically equal to those of the pullback of the Gaussian free field on the upper half plane. As far as we know, the results are new even for Poisson/Bernoulli nonintersecting random walks.

在Bufetov和Gorin撰写的一系列论文[23]，[24]，[25]中，舒尔产生的函数是as单元上的傅立叶变换 $ü（ñ）$引入来研究随机N粒子系统的渐近行为。我们介绍和研究随机N粒子系统的Jack生成函数。在特殊情况下，这可以看作是对Gelfand对的傅立叶变换$（G{\mathrm{大号}}_{ñ}（\mathbb{[R}），Ø（ñ））$， $（G{\mathrm{大号}}_{ñ}（\mathbb{C}），ü（ñ））$ 和 $（G{\mathrm{大号}}_{ñ}（\mathbb{H}），\mathrm{小号}p（ñ））$。我们的主要结果表明，此类粒子系统的大数定律和中心极限定理，等于其杰克生成功能统一的细菌某些条件。我们的结果提供了一个新的工具箱，用于通过其Jack生成函数来分析随机离散粒子系统的整体行为，以及通过随机离散动力学来分析Jack对称函数的极限行为。我们主要结果的一些应用包括：

1。

利用大数定律和β非相交泊松随机游动的中心极限定理[33]作为输入，我们的定理给出了杰克字符的渐近性。

2。

我们证明了区域多项式的Littlewood-Richardson系数的大数定律和中心极限定理。

3。

我们表明，不相交的随机游走的一般族的高度函数的波动渐近地等于上半平面上的高斯自由场回撤的高度。据我们所知，即使对于泊松/伯努利不相交的随机游走，结果也是新的。