Topics in the Spectral Theory of Random Operators and in Statistical Mechanics

随机算子谱理论和统计力学主题

• 批准号: 1305472
• 负责人: Michael Aizenman
• 金额: $ 18.6万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1305472&HistoricalAwards=false
• 关键词: Topics; Spectral; Theory; Random; Operators

This research focusses on two groups of topics in mathematical physics: the spectral theory of random operators, and statistical mechanics models, with special focus on two and quasi-one-dimensional systems. A wide range of methods from mathematical physics, classical analysis, probability theory, and convex geometry will be applied to study some of the open problems in these areas. In addition, the connections between the two areas will be investigated. Several major problems pertaining to random band operators remain open after many years of research. Perturbative series (such as resolvent expansions) typically diverge in the physically interesting regimes, whereas the non-perturbative methods (supersymmetric formalism, transfer operators) have been only partially developed. The PI will develop perturbative and non-perturbative methods to improve the rigorous understanding of the spectral properties of random band operators. In two- and higher dimensional statistical mechanics, the PI will investigate the large scale properties of gradient models, as well as discrete models. A better understanding of the former will also shed light on the latter (for example, on two-dimensional height models). The PI will strive to combine the well-developed multi-scale and convexity methods with techniques from analysis and high-dimensional convex geometry. Broader impacts This project will develop the connections between mathematical physics and other fields of mathematics, especially, probability theory, classical analysis and convex geometry. These connections may lead to developments in all these fields. The PI has delivered numerous talks at seminars, colloquia, and international conferences in mathematical physics, analysis, probability, high-dimensional geometry, and the interactions between these areas.

这项研究的重点是数学物理学中的两组主题：随机操作者的光谱理论和统计力学模型，特别关注两个和准二维系统。数学物理学，经典分析，概率理论和凸几何形状的广泛方法将用于研究这些领域的一些开放问题。此外，将研究两个区域之间的连接。经过多年的研究，与随机乐队运营商有关的几个主要问题仍然开放。扰动序列（例如分解膨胀）通常在物理上有趣的方案中有所不同，而非扰动方法（超对称形式，转移算子）仅部分开发出来。 PI将开发扰动和非扰动方法，以提高对随机带算子光谱特性的严格理解。在两维统计力学中，PI将研究梯度模型的大规模特性以及离散模型。对前者的更好理解也将揭示后者（例如，在二维高度模型上）。 PI将努力将发达的多尺度和凸方法与分析和高维凸几何形状相结合。更广泛的影响该项目将发展数学物理学和其他数学领域之间的联系，尤其是概率理论，经典分析和凸几何。这些连接可能会导致所有这些领域的发展。 PI在数学物理学，分析，概率，高维几何形状以及这些领域之间的数学物理学，分析，概率，高维几何形状以及这些领域之间的相互作用方面进行了众多演讲。