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Partial isometries, duality, and determinantal point processes
Random Matrices: Theory and Applications ( IF 0.9 ) Pub Date : 2021-10-22
Makoto Katori, Tomoyuki Shirai

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, H,=1,2, which are assumed to be realized as L2-spaces, L2(S,λ), =1,2, and introduce a bounded linear operator 𝒲:H1H2 and its adjoint 𝒲:H2H1. 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 𝒲𝒲. In addition, if 𝒲 is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, (Ξ,K,λ), =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) series of infinite DPPs on S=d and d 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 在一个空间 有量度 λ,其相关函数均由积分核指定的行列式给出 称为相关核。我们考虑一对希尔伯特空间,H,=1,2,假设被实现为 2- 空格, 2(,λ), =1,2,并引入一个有界线性算子 𝒲H1H2 和它的伴随 𝒲H2H1. 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 (Ξ1,1,λ1) 有关联 𝒲𝒲. 此外,如果𝒲 也是局部 Hilbert-Schmidt 类,那么我们有一对独特的 DPP, (Ξ,,λ), =1,2. 我们还提供了一个实用的框架,使𝒲𝒲满足以上条件。我们构建 DPP 对的框架暗示了 DPP 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种不同的表达方式,揭示 DPP 的不同方面。为了证明我们框架的这些优点,并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,我们报告了大量的单、双和 DPP 示例。高维空间,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) 上的无限 DPP 系列 =dd 根据 Zelditch 的术语,我们分别讨论了 DPP 的欧几里得家族和海森堡家族。

更新日期:2021-10-25
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