A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures Ξ on a space S with measure λ, whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, Hℓ,ℓ = 1, 2, which are assumed to be realized as L2-spaces, L2(S ℓ,λℓ), ℓ = 1, 2, and introduce a bounded linear operator 𝒲 : H1 → H2 and its adjoint 𝒲∗ : H 2 → H1. We show that if 𝒲 is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP (Ξ1,K1,λ1) associated with 𝒲∗𝒲. In addition, if 𝒲∗ is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, (Ξℓ,Kℓ,λℓ), ℓ = 1, 2. We also give a practical framework which makes 𝒲 and 𝒲∗ satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter (d ∈ ℕ) series of infinite DPPs on S = ℝd and ℂd are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

中文翻译：

---

部分等距、对偶和行列式点过程

行列式点过程 (DPP) 是随机非负整数值氡测量的集合Ξ在空间上小号有措施λ，其相关函数均由积分核指定的行列式给出ķ称为相关核。我们考虑一对希尔伯特空间，Hℓ,ℓ = 1, 2, 假设被实现为大号2-空格，大号2(小号 ℓ,λℓ),ℓ = 1, 2，并引入有界线性算子𝒲 ： H1 → H2及其伴随𝒲* ： H 2 → H1. 我们证明如果𝒲是局部 Hilbert-Schmidt 类的部分等距，那么我们有一个唯一的 DPP(Ξ1,ķ1,λ1)有关联𝒲*𝒲. 此外，如果𝒲*也是局部 Hilbert-Schmidt 类，那么我们有一对独特的 DPP，(Ξℓ,ķℓ,λℓ),ℓ = 1, 2. 我们还给出了一个实用的框架，使𝒲和𝒲*满足以上条件。我们构建 DPP 对的框架暗示了 DPP 配对之间有用的二元关系。对于给定 DPP 的相关核，我们的公式可以提供多个不同的表达式，这些表达式揭示了 DPP 的不同方面。为了证明我们框架的这些优势以及表明通过这种方法获得的 DPP 类足够大，可以研究各种 DPP 中的通用结构，我们报告了大量的一、二和高维空间小号，其中给出了从有限 DPP 到无限 DPP 的几种类型的弱收敛。一参数（d ∈ ℕ) 一系列无限 DPP小号 = ℝd和ℂd我们分别按照 Zelditch 的术语将其称为 DPP 的欧几里得家族和海森堡家族。