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

LEAPS-MPS：自由概率和随机矩阵理论的一些应用

• 批准号: 2316836
• 负责人: Ping Zhong
• 金额: $ 24.75万
• 依托单位: University of Wyoming
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316836&HistoricalAwards=false
• 关键词: LEAPS; MPS; Some; Applications; Free; LEAPS; MPS; Some; Applications; Free

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.

具有随机条目的矩阵在物理，统计和工程上自然而然，它们旨在描述复杂的系统。由于这些物品的尺寸通常很大，因此线性代数中的经典工具不足以应对这种情况。随机矩阵理论的一个共同主题是，由于普遍性现象，大型随机物质家族具有相同的限制分布。这是经典概率中核心限制定理的类似物，在经典概率上，i.i.d的唯一要求。随机变量是瞬间条件。因此，在非常温和的假设下，随机矩阵理论可以理解大规模数据。较大维度的随机物质通常可以由生活在某些抽象运营商代数中的非随机操作员建模，在该代数中，这些操作员满足了Voiculescu自由独立性的一些高度非平凡的关系。这些非随机操作员是自由概率理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。该项目得到了LEAPS-MPS奖的支持，旨在开发分析工具，以研究有关重要随机矩阵模型的限制分布的基本问题。这些问题是由数学，统计，组合和量子信息的问题引起的。游离随机变量的棕色测量是一种光谱测量，可概括平方材料的特征值分布。一个主要目的是开发用于计算棕色测量结果的新技术，该技术为非官员随机物质的限制提供了预测。 Hermitian还原方法和从属功能是推导棕色测量公式的强大工具。棕色测量结果的新结果打开了研究以前无法访问的随机基质模型的大门。自由概率理论提供了一种概念性的方法来研究大维的随机矩阵。主要研究者将确定由高维统计和量子信息理论引起的各种随机矩阵模型的限制自由随机变量。特别是，主要研究者将检查完整秩的单圈随机矩阵模型，时间序列的自动增强矩阵矩阵和随机张量网络的K阳性。此外，PI将探索epsilon柔性理论，并研究其在量子信息中的应用。该项目部分由NSF建立的计划刺激竞争研究（EPSCOR）资助。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响评估标准来通过评估来获得的支持。