LEAPS-MPS: Some Applications of Free Probability and Random Matrix Theory

具有随机元素的矩阵在物理学、统计学和工程学中自然出现，它们被设计用来描述复杂的系统。由于这些矩阵的维数通常非常大，线性代数中的经典工具不足以解决这种情况。随机矩阵理论的一个共同主题是，由于普适性现象，一大群随机矩阵共享相同的极限分布。这是一个类似的中心极限定理在经典概率，其中唯一的要求，独立同分布。随机变量是一些矩条件。因此，随机矩阵理论可以在非常温和的假设下解释大规模数据。大维数的随机矩阵通常可以用抽象算子代数中的非随机算子来模拟，这些算子满足Voiculescu自由独立的高度非平凡关系。这些非随机算子是自由概率论中的自由随机变量。主要研究者将研究自由随机变量的概率分布和合适的随机矩阵模型的收敛性。该项目为本科生和研究生提供了研究机会。该项目由LEAPS-MPS奖支持，旨在开发用于研究重要随机矩阵模型极限分布基本问题的分析工具。这些问题的动机来自数学、统计学、组合学和量子信息。自由随机变量的布朗测度是一种谱测度，它推广了方阵的特征值分布。一个主要目标是开发新的技术来计算布朗措施，提供预测的限制，非厄米特随机矩阵。厄米特约化方法和隶属函数是导出布朗测度公式的有力工具。布朗措施的新结果打开了大门，以前无法访问的随机矩阵模型的研究。自由概率论提供了一个概念性的方法来研究随机矩阵的大尺寸。主要研究者将确定从高维统计和量子信息理论产生的各种随机矩阵模型的限制自由随机变量。特别是，首席研究员将研究满秩变形单环随机矩阵模型的谱，时间序列的自协方差矩阵和随机张量网络的k-正性。此外，PI将探索ε-freeness理论并研究其在量子信息中的应用。 该项目部分由NSF激励竞争研究既定计划（EPSCoR）资助。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Matrices with random entries arise naturally in physics, statistics, and engineering, and they are designed to describe complicated systems. As the dimensions of these matrices are usually very large, classical tools in linear algebra are inadequate to tackle this situation. One common theme of random matrix theory is that a large family of random matrices shares the same limiting distribution due to universality phenomena. This is an analogue of the central limit theorem in classical probability, where the only requirements for the i.i.d. random variables are some moment conditions. Hence, random matrix theory can make sense of large-scale data under very mild assumptions. Random matrices of large dimension can often be modeled by nonrandom operators living in some abstract operator algebras, where these operators satisfy some highly nontrivial relations characterized by Voiculescu’s free independence. These nonrandom operators are free random variables in free probability theory. The principal investigator will study probability distributions of free random variables and the convergence of suitable random matrix models. The project provides research opportunities for both undergraduate and graduate students. This project, supported by a LEAPS-MPS award, aims to develop analytic tools for studying fundamental questions regarding the limiting distributions of important random matrix models. These questions are motivated by questions from mathematics, statistics, combinatorics, and quantum information. The Brown measure of a free random variable is a spectral measure that generalizes the eigenvalue distribution of square matrices. One major objective is to develop new techniques for calculating Brown measures, which provide predictions for the limits of non-Hermitian random matrices. The Hermitian reduction method and subordination functions are powerful tools for deriving Brown measure formulas. The new results on Brown measures open the door to the study of random matrix models that were previously inaccessible. Free probability theory offers a conceptual approach to studying random matrices of large dimensions. The principal investigator will identify the limiting free random variables for various random matrix models arising from high-dimensional statistics and quantum information theory. In particular, the principal investigator will examine the spectrum of the full rank deformed single-ring random matrix model, the autocovariance matrix of time series, and the k-positivity of random tensor networks. Additionally, the PI will explore the theory of epsilon-freeness and investigate its applications to quantum information. This project is funded in part by the NSF Established Program to Stimulate Competitive Research (EPSCoR).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.