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

提出的研究项目集中于研究随机介质中的随机矩阵和随机游动。现有的许多随机矩阵理论都涉及正规矩阵，它们在扰动下是稳定的；许多现有工具简单地使用这种稳定性，或者一些强独立假设，甚至在推导粗略结果时也是如此。本提案的一个主要目标是在非政府组织和合作者以前工作的基础上，开发即使在结构假设(常态或独立性)都不成立的情况下也能发挥作用的技术。虽然该提案讨论和解决了一些非常具体的问题，但它是开发适用于广泛RMT问题的新技术的更大尝试的一部分。第二个要研究的课题是随机环境中的随机游动和分支过程。尽管随机行走的理论已经发展得很好，但当人们将行走演变为随机介质的媒介改变时，理解上的许多差距仍然存在。尽管一些研究人员在过去几年里取得了快速进展，但许多基本问题仍然没有得到回答。这个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件。随机矩阵理论(RMT)研究随机矩阵的性质，特别是在矩阵的维度很大的极限内。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(特别是通信和信息理论)的多个领域。近年来，RMT已经成为数学中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。最近已经解决了几个主要的(和基本的)公开问题，特别是关于大型随机矩阵谱的极限定律的普适性。这项拟议的研究将寻求将这一理论显著扩展到一类在(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，例如在评估复杂系统的稳定性方面，在与量子信息理论相关的计算中，在统计学方面。这项研究的第二个主要焦点，随机游动，可以说是数学家研究最多的随机过程，在物理科学、工程和社会科学等领域有着最广泛的应用。虽然随机游动的理论到目前为止已经发展得很好，但当人们将游动演化到随机介质的介质改变为随机介质，从而获得随机环境中的随机游动(RWRE)时，情况并非如此。这种RWRE可以用来模拟物理和工程科学中的许多随机介质中的运动问题，在数学上很有吸引力，一方面是因为该模型非常简单，另一方面，已经建立的研究随机介质中过程的工具不适用于RWRE的研究。当前提案的目标是开发新的基本概率技术，使之能够对RWRE的行为做出可证明的预测。预计这种技术将在其他过程的研究中有用，并自然地与陷阱模型和增强的随机游动的研究相联系。虽然在这项提案中没有明确的目标，但诱捕模型最近被用于研究核废物的环境影响，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污染在真实环境中的扩散。

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.