与Heisenberg群相关的行列式点过程的超一致性
Hyperuniformity of the determinantal point processes associated with the Heisenberg group
摘要：
Ginibre点过程是由非厄米复高斯矩阵在无限大矩阵尺寸极限下的特征值分布给出的。这是复平面${\mathbb{C}}$上的一个决定点过程（DPP），因为所有的相关函数都是由称为相关核的积分核所指定的决定因素给出的。Shirai介绍了Ginibre DPP的单参数（$m \in {\mathbb{N}}_0$）扩展，并称之为Ginibre型点过程。本文考虑${\mathbb{C}}$上的Ginibre点过程和Ginibre型点过程推广到高维空间${\mathbb{C}}^D, D=2,3, \dots$中的DPP，其中DPP由多元水平$m \in {\mathbb{N}}_0^D$参数化.我们称所得到的点过程为扩展的海森堡DPP族，因为相关核通常等同于由薛定谔表示表示的海森堡群空间中两点的相关。我们证明了这个大家族中的所有DPP都是超均匀的I类。
英文摘要：
The Ginibre point process is given by the eigenvalue distribution of a non-hermitian complex Gaussian matrix in the infinite matrix-size limit. This is a determinantal point process (DPP) on the complex plane ${\mathbb{C}}$ in the sense that all correlation functions are given by determinants specified by an integral kernel called the correlation kernel. Shirai introduced the one-parameter ($m \in {\mathbb{N}}_0$) extensions of the Ginibre DPP and called them the Ginibre-type point processes. In the present paper we consider a generalization of the Ginibre and the Ginibre-type point processes on ${\mathbb{C}}$ to the DPPs in the higher-dimensional spaces, ${\mathbb{C}}^D, D=2,3, \dots$, in which they are parameterized by a multivariate level $m \in {\mathbb{N}}_0^D$. We call the obtained point processes the extended Heisenberg family of DPPs, since the correlation kernels are generally identified with the correlations of two points in the space of Heisenberg group expressed by the Schrödinger representations. We prove that all DPPs in this large family are in Class I of hyperuniformity.
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