CAREER: Analysis of disordered systems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目将侧重于对无序介质中波传播的严格分析，以及对随机算子和矩阵的相关分析。目标是建立波在弱无序介质中任意长时间的扩散传播。在物理文献中有丰富的非严格的波扩散理论，基于启发式分析和不受控制的微扰理论，表明扩散发生并预测了扩散常数的阶渐近行为。尽管这个问题显然很重要，但我们还远远没有对所涉及的数学问题进行严格的分析。PI将从多个角度考虑这个问题，特别是使用“增强空间”表示，例如在随机介质和均匀化中的随机游走分析中被证明是有用的。此外，PI将考虑随机矩阵理论中的相关问题，特别是随机带矩阵，其条目在对角线周围的带外消失，这被建议作为所谓的金属-绝缘体过渡模型，因为带宽从矩阵大小移动到1。紊乱的影响是什么？这是一个关于任何物理模型的基本问题，即使是没有明确包括无序的模型。毕竟，任何真实世界的系统都受到少量噪声的影响，经验表明，即使是微弱的无序也可能对系统的行为产生深远的影响。尽管这门学科具有基本性质，但人们对它的各个方面仍然知之甚少，目前还没有以一种容易理解的方式纳入课程。本项目旨在通过以下途径弥合这一差距：1)为本科生开设统计力学和随机矩阵理论的基本模型课程；2)集中研究与波动和半导体物理相关的问题。这项工作的目标是使用分析工具来进一步了解理论物理和应用数学的模型。这些进步在短期内影响了数学和理论物理以外的领域，产生了构建新的、更好的应用模型的想法。然而，从长远来看，影响将会更大，因为数学工作阐明了物理理论中那些真正基础的部分。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The project will focus on a rigorous analysis of wave propagation in disordered media, along with related analysis of random operators and matrices. The goal is to establish diffusive propagation of waves in a weakly disordered medium over arbitrarily long times. There is a rich non-rigorous theory of wave diffusion in the physics literature, based on heuristic analyses and uncontrolled perturbation theory, which suggests that diffusion occurs and predicts the leading order asymptotic behavior of the diffusion constant. In spite of the obvious importance of this problem, we are far from having a rigorous analysis of the mathematics involved. The PI will consider the problem from a number of view points, in particular using an ``augmented space" representation such as has proved useful in the analysis of random walks in random media and homogenization. Additionally the PI will consider related problems in the theory of random matrices, specifically random band matrices, with entries that vanish outside a band around the diagonal, which have been suggested as a model of a so-called metal-insulator transition as the bandwidth is moved from the matrix size down to one.What are the effects of disorder? This is a fundamental question in regards to any model of physics, even one in which disorder is not explicitly included. After all, any real world system is subject to a small amount of noise, and experience shows that even weak disorder may have a profound effect on the behavior of the system. Despite the fundamental nature of this subject, aspects of it remain poorly understood and it is not at present incorporated in the curriculum in an accessible way. This program seeks to bridge this gap as follows: 1) through the development of courses for undergraduates on the basic models of statistical mechanics and random matrix theory; and 2) by focused research on aspects of the question relevant to wave motion and semi-conductor physics. The goal of such work is to use analytical tools to further basic understanding of models of theoretical physics and applied mathematics. Such advances impact areas outside of mathematics and theoretical physics in the short run by yielding ideas for constructing new and better models useful in applications. In the long run, however, the impact will be even greater, as mathematical work illuminates those parts of physical theory that are truly fundamental.