Invariant Ensembles of Random Matrices: New Techniques, New Horizons

当代的随机矩阵理论已经成为分析由许多高度相关的组件组成的复杂随机系统的一个可行的模型框架。这样的系统在数学和自然界中无处不在，从复数论函数到原子核，再到公共交通系统。这个项目将开发一种新的技术，用于对一大类随机矩阵进行频谱分析，这些随机矩阵的分布不受连续对称群的作用的影响。这个项目的目标是将代数组合学和几何(Weingarten演算，Hurwitz理论)的新方法引入到随机矩阵的不变系综的研究中，开发一种非常通用和健壮的新方法来研究这些系综的谱理论。具体地说，该项目的目标是开发一种类似于Wigner矩方法的方法，该方法专门针对不变系综群。这种方法将被用来分析随机矩阵的不变系综谱统计，在单矩阵和更一般的多矩阵情况下都是如此。所发展的矩方法不限于具有绝对连续律的随机矩阵，特别是它适用于轨道测量的离散混合。这导致了随机矩阵理论和统计机制之间重要的新联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The contemporary subject of Random Matrix Theory has emerged as a viable model framework for the analysis of complex random systems comprised of many highly correlated components. Such systems are ubiquitous in mathematics and nature, ranging from complex number-theoretic functions, to atomic nuclei, to public transit systems. This project will develop new techniques for the spectral analysis of a broad class of random matrices whose distribution is immune to the action of a continuous group of symmetries. The novel approach which will be developed has the potential to dramatically expand our current understanding of this important paradigm.The goal of this project is to bring new methods from algebraic combinatorics and geometry (Weingarten calculus, Hurwitz theory) to the study of invariant ensembles of random matrices, developing a new approach to the spectral theory of these ensembles which is very general and robust. Specifically, the goal of the project is to develop an analogue of Wigner's moment method which specifically targets invariant ensembles. This approach will be used to analyze spectral statistics of invariant ensembles of random matrices, in both the single-matrix and more general multi-matrix cases. The moment method developed is not limited to random matrices with an absolutely continuous law; in particular, it is applicable to discrete mixtures of orbital measures. This leads to important new connections between random matrix theory and statistical mechanics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.