高维复空间中单位球内共形不变行列式点过程与泊松点过程

The theory of determinantal point processes arises from mathematical physics, which is the mathematical model for the random fermion fields. This theory has interactions with limit theory in probabilities, random matrix theory, KPZ equations, complex analysis, functional analysis, ergodic theory, random graph theory, etc. The theory of Poisson point processes is the oldest theory in point processes and is widely used in pure and applied mathematics. ..This project aims to investigate a class of conformally invariant determinantal and Poisson point processes in the unit ball of higher dimensional complex spaces. The main goals are: 1) to study when a conformally invariant Poisson point process is almost surely the uniqueness set of the weighted Bergman spaces；2) to investigate the re-construction of a square-integrable holomorphic Bergman function from its restriction on a uniqueness set; 3) to study when a conformally invariant point process is almost surely the uniqueness set of the weighted harmonic Bergman spaces; 4) to study the Pfaffian point processes related to Bergman kernels on the unit balls; 5) to obtain the classification of conformally invariant determinantal point processes on the unit balls of higher dimensional complex spaces.

行列式点过程理论的研究涉及到数学物理、概率中极限理论、随机矩阵、KPZ方程、复分析、泛函分析、遍历论、随机图论等多个方向，是当前国际上广受关注的新兴和热点领域。行列式点过程理论主要研究各种度量空间中具有特殊排斥作用的一类随机离散集的性质，该理论起源于数学物理中的随机费米子场。行列式点过程比概率论中最广为应用的泊松点过程更为复杂。..本项目计划研究一类在高维复空间单位开球内共形不变的行列式点过程与泊松点过程的极限性质，主要内容包括：1）共形不变泊松点过程的Lyons完备性问题；2）Bergman解析函数的重构问题；3）Bergman调和函数空间的唯一集问题；4）高维单位球内Pfaffian点过程问题；5）高维单位球内共形不变行列式点过程的分类问题。

行列式点过程理论的研究涉及到数学物理、概率中极限理论、随机矩阵、KPZ方程、复分析、泛函分析、遍历论、随机图论等多个方向，是当前国际上广受关注的新兴和热点领域。本项目主要研究高维复空间单位开球内共形不变的行列式点过程与泊松点过程的极限性质以及与点过程、平稳过程等相关的其他研究问题。目前取得的成果包括1）构造了单位圆盘上一类具有点刚性的行列式点过程；2）给出并证明了任意维单位复球上由Bergman核诱导的行列式点过程对于加权Bergman空间情形，Hardy型空间等的Patterson-Sullivan型重构公式; 3) 构造了一类分枝型平稳过程并给出齐次树指标情形下该类过程的谱测度的完全分类。