Distributions on partitions, point processes, and the hypergeometric kernel
Distributions on partitions, point processes, and the hypergeometric kernel
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DOI：
10.1007/s002200050815
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发表时间：
2000-04-01
影响因子：
2.4
通讯作者：
Olshanski, G
中科院分区：
文献类型：
--
作者：
Borodin, A;Olshanski, G
我们研究一维晶格上的随机点过程的 3 参数族，该过程源自无限对称群的一个显着的表示族。我们证明了过程的相关函数是由具有一定核的行列式给出的。核可以通过高斯超几何函数来表示；我们称之为超几何内核。在缩放限制下，我们的过程近似于描述将上述表示分解为不可约的过程。正如我们在之前的工作中所展示的，这些极限过程的相关函数也具有所谓的 Whittaker 核的行列式形式。我们证明超几何核的标度极限是Whittaker核。Whittaker核对应的积分算子是由Its、Izergin、Korepin和Slavnov定义的可积算子。我们认为超几何核可以被认为是定义“离散可积算子”的核。我们还表明，对于 Meixner 正交多项式，超几何核对于 Christoffel-Darboux 核的某些参数值会退化。这一事实与拉盖尔多项式的 Whittaker 核退化为 Christoffel-Darboux 核是平行的。
We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel.In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above into irreducibles. As we showed in previous works, the correlation functions of these limit processes also have determinantal form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel.The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered as a kernel defining a 'discrete integrable operator'.We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel-Darboux kernel for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel-Darboux kernel for Laguerre polynomials.