行列式点过程的概率分析

Determinantal point processes are a special class of random point processes with elegant algebraic structure, whose correlation functions can be written as a determinant of a kernel function. The determinantal point processes appear widely in various probability problems such as random matrices, random partitions, random growth models, random combinatorial optimization. It has become a very popular and hot research topic in probability theory and particularly probability limit theory in recent years. The present project is devoted to the study of probability limit theory for determinantal point processes and its applications. The content mainly consists of : (1) Discovering some new random point processes (problemsor models) with determinantal correlation functions, and characterizing the kernel function structure to anaylze; (2) studying the convergence for various statistics of point processes like counting measures, linear sums, extremal points, the gap between points, and describing properties of their limit laws; (3) studying the convergence of discrete point processes after properly normalized and describing the limit processes using stochastic calculus; (4) further developing the ideas of determinantal point processes and applying to other more complicated and harder point processes, say deformed versions of classical random matrix models and random partitions. Through this project, we will be able to the inner law of a big class of probability problems, and hencefore richen and push forward the development of probability limit theory.

行列式点过程是一类具有良好代数结构的特殊随机点过程，其相关函数可以用核函数的行列式表示。它广泛地出现在随机矩阵、随机划分、随机增长模型、随机组合优化等问题中，近年来成为概率论学科、特别是概率极限理论领域的研究热点。 本项目致力于行列式点过程的概率极限理论及其应用研究。内容包括：（1）探索具有行列式相关函数的新型点过程（问题、模型）， 刻画核函数结构并加以分析；（2）研究行列式点过程的各种统计量， 如个数、线性和、极值点、点间隙等的极限分布， 刻画所得极限分布的渐近性质；（3）研究离散化点过程经过适当规范化后的收敛性及其极限过程，运用随机分析方法刻画极限过程的分布规律；（4）进一步发展行列式点过程的思想，并用于研究其他更为复杂、更为困难的点过程， 如经典随机矩阵模型、随机划分过程的变化形式。 通过本项目的实施， 可以揭示一大类概率问题的内在分布规律，丰富和发展概率极限理论研究。

行列式点过程具有良好代数结构，其相关函数可以用核函数的行列式表示. 它广泛地出现在随机矩阵、随机划分、随机增长模型等问题中，近年来成为概率极限理论领域的研究热点...本项目着重研究.(1).随机整数划分在Plancherel 测度和乘积测度下的渐近分布理论；.(2).随机混合有向聚合模型的能量流体动力学极限和自由能量波动； .(3).高维样本协方差矩阵特征根点过程的渐近分布及在高维统计推断中的应用；.(4).随机点过程随机积分的弱收敛性和大偏差不等式..代表性成果如下.[1] 《Random Matrices and Random Partitions—Normal Convergence 》一书2015年由World Scientific Publish出版, 全书特色鲜明，内容丰富，结构清晰，体系完整.[2] 《Probabilistic Analysis of Random Integer Partitions》系统地总结了整数划分在各种概率测度下的极限形和波动分布. 此文被推荐刊登在《数学进展》，纪念该刊创立60周年.[3]《Tracy-Widom分布及其应用》系统介绍了Tracy-Widom分布的发现历史，密度函数表示特点，以及在概率论、统计物理、数理统计、组合分析等学科中的应用.[4] 《Free Energy Fluctuations for a Mixture of Directed Polymers ( Science China: Math.)》研究有向聚合模型的混合，在一定的尺度变换下，建立自由能量的流体动力极限和Tracy–Widom 渐近分布，给出极限常数和分布的方差.[5]《Spectral Statistics of Large Dimensional Spearman’s Rank Correlation Matrix and its Applications (Ann.Stat.)》建立Spearman 秩相关系数矩阵谱统计量的渐近分布， 提出“两步比较方法”给出均值和协方差矩阵的详细估计公式.[6]《On Bernstein Type Inequality for Stochastic Integrals of Multivariate Point Processes(待SPA)》研究随机积分，获得Bernstein 型大偏差