Study of random motion in random environment, and random matrix theory

随机环境中的随机运动和随机矩阵理论的研究

• 批准号: 1203201
• 负责人: Maury Bramson
• 金额: $ 40.05万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2018-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1203201&HistoricalAwards=false
• 关键词: Study; random; motion; environment; matrix

The proposed research project is focused on the study of random matrices and random walks evolving in random media. Much of the existing theory of random matrices deals with normal matrices, which are stable under perturbations; many of the existing toolsimplicitly use either this stability, or some strong independence assumptions, even in deriving rough results. A major goal of the present proposal is to build on previous work of the PI and collaborators and develop techniques that work even in a context where both structural assumptions (normality or independence) fail. While the proposal discusses and addresses some very specific questions, it is part of a larger attempt to develop new techniques that would be applicable to a wide range of RMT questions. The second topic to be studied concerns random walks and branching processes in random environments. Though the theory of random walks is well developed, many gaps in understanding remain when one changes the medium in which the walk evolves to a random medium. In spite of rapid progress that was achieved in the last few years by several researchers, many fundamental questions remain unanswered. The proposed project will built on previous work, by the PI and other researchers, with the goal of resolving some of these outstanding questions.Matrices are the cornerstone of linear algebra, and are fundamental building blocks in operator theory. Random matrix theory (RMT) is concerned with the study of properties of random matrices, typically in the limit where the dimension of the matrix is large.Motivations and applications for this study come from several areas of mathematics (probability theory, number theory, representation theory), the physical sciences (especially, mathematical physics), statistics, and engineering (specifically, communication and information theories). In recent years, RMT has emerged as a major research area within mathematics, combining techniques from probability theory, operator algebras, complex analysis, and combinatorics. Several major (and fundamental) open questions have recently been resolved, especially concerning the universality of limit laws regarding the spectrum of large random matrices. The proposed research will seek to significantly expand the theory toward a class of matrices whose spectrum is not stable under (small) perturbations. While the focus of the research is theoretical, an impact on applications is expected, e.g. in evaluating the stability of complex systems, in computations related to quantum information theory, and in statistics. The second main focus of the study, random walks, are arguably the stochastic processes most studied by mathematicians, having the widest range of applications in fields as diverse as the physical sciences, engineering, and the social sciences. Though the theory of random walks is by now well developed, this is not at all the case when one changes the medium in which the walk evolves to a random medium, thus obtaining a random walk in random environment (RWRE). Such RWRE's can be used to model many problems of motion in random media in the physical and engineering sciences, and are mathematically appealing because on the one hand the model is very simply stated, while on the other hand established tools for studying processes in random media are not applicable in the study of RWRE. The goal of the current proposal is to develop new basic probabilistic techniques that will allow to make provable predictions concerning the behavior of RWRE. It is expected that such techniques will be useful in the study of other processes, and link naturally to the study of trapping models and reinforced random walks. While not explicitly targeted in this proposal, trapping models have recently been used to study the environmental impact of nuclear waste, and RWRE's can naturally be used in this context to model the spread of contamination in a real environment, once the required mathematical background and tools are in place.

拟议的研究项目侧重于研究随机矩阵和随机介质中演化的随机游走。许多现有的随机矩阵理论都涉及正态矩阵，这些矩阵在扰动下是稳定的。许多现有工具隐含地使用这种稳定性或一些强独立性假设，即使在得出粗略结果时也是如此。本提案的一个主要目标是建立在 PI 和合作者之前的工作基础上，开发即使在结构假设（正态性或独立性）都失败的情况下也能发挥作用的技术。虽然该提案讨论并解决了一些非常具体的问题，但它是开发适用于广泛的 RMT 问题的新技术的更大尝试的一部分。要研究的第二个主题涉及随机环境中的随机游走和分支过程。尽管随机游走理论已经很成熟，但当人们将游走演化为随机介质的媒介改变时，在理解上仍然存在许多差距。尽管一些研究人员在过去几年中取得了快速进展，但许多基本问题仍未得到解答。拟议的项目将建立在 PI 和其他研究人员之前的工作基础上，目标是解决其中一些悬而未决的问题。矩阵是线性代数的基石，也是算子理论的基本构建模块。随机矩阵理论（RMT）涉及随机矩阵性质的研究，通常是在矩阵维数较大的情况下进行的。这项研究的动机和应用来自数学（概率论、数论、表示论）、物理科学（特别是数学物理）、统计学和工程学（特别是通信和信息论）的多个领域。近年来，RMT 已成为数学领域的一个主要研究领域，它结合了概率论、算子代数、复分析和组合学的技术。 最近解决了几个主要（和基本）的开放问题，特别是关于大型随机矩阵谱的极限定律的普遍性。拟议的研究将寻求将该理论显着扩展到一类在（小）扰动下频谱不稳定的矩阵。虽然研究的重点是理论上的，但预计会对应用产生影响，例如评估复杂系统的稳定性、与量子信息理论相关的计算以及统计学。该研究的第二个主要焦点是随机游走，它可以说是数学家研究最多的随机过程，在物理科学、工程和社会科学等不同领域具有最广泛的应用。虽然随机游走的理论现在已经很发达，但是当我们将游走演化的介质改变为随机介质，从而获得随机环境中的随机游走（RWRE）时，情况就完全不是这样了。这种 RWRE 可用于模拟物理和工程科学中随​​机介质中的许多运动问题，并且在数学上很有吸引力，因为一方面该模型非常简单，而另一方面用于研究随机介质中的过程的既定工具不适用于 RWRE 的研究。当前提案的目标是开发新的基本概率技术，以便对 RWRE 的行为做出可证明的预测。预计此类技术将在其他过程的研究中有用，并自然地与捕获模型和强化随机游走的研究联系起来。虽然该提案没有明确针对的目标，但捕获模型最近已被用于研究核废料对环境的影响，一旦所需的数学背景和工具到位，RWRE 自然可以在这种情况下用于模拟真实环境中的污染扩散。