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。该项目将专注于对无序介质中波传播的严格分析，沿着随机算子和矩阵的相关分析。 我们的目标是建立一个弱无序介质中任意长时间的波的扩散传播。 在物理学文献中有丰富的波扩散的非严格理论，基于启发式分析和非受控扰动理论，其表明扩散发生并预测扩散常数的领先阶渐近行为。 尽管这个问题的重要性显而易见，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”表示法，例如在随机介质和均匀化中的随机游走分析中已被证明是有用的。 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，其元素在对角线周围的带外消失，这被认为是所谓的金属-绝缘体转变的模型，因为带宽从矩阵大小下降到1。无序的影响是什么？ 这是任何物理模型的基本问题，即使是没有明确包括无序的物理模型。 毕竟，任何真实的世界系统都会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。 尽管这一学科具有根本性，但人们对它的某些方面仍然知之甚少，而且目前还没有以一种易于理解的方式将其纳入课程。该计划旨在弥合这一差距如下：1）通过统计力学和随机矩阵理论的基本模型的本科生课程的发展;和2）通过集中研究有关波动和半导体物理问题的方面。 这项工作的目标是使用分析工具，进一步了解理论物理和应用数学的模型。 这些进展在短期内影响了数学和理论物理之外的领域，为构建新的和更好的模型提供了新的想法。然而，从长远来看，影响会更大，因为数学工作阐明了物理理论中真正基本的部分。

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.