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

本研究集中在数学物理的两组主题：随机算子的谱理论和统计力学模型，特别关注二维和准一维系统。从数学物理，经典分析，概率论和凸几何的方法将广泛应用于研究这些领域的一些开放的问题。此外，还将研究这两个领域之间的联系。经过多年的研究，与随机频带算子有关的几个主要问题仍然没有解决。微扰级数（如预解式展开）通常在物理上感兴趣的区域发散，而非微扰方法（超对称形式主义，转移算子）只得到了部分发展。PI将开发微扰和非微扰方法，以提高对随机带算子谱特性的严格理解。在二维和高维统计力学中，PI将研究梯度模型以及离散模型的大尺度特性。更好地理解前者也将有助于理解后者（例如，二维高度模型）。PI将努力将成熟的多尺度和凸性方法与分析和高维凸几何技术相结合。该项目将发展数学物理与其他数学领域之间的联系，特别是概率论，经典分析和凸几何。这些联系可能导致所有这些领域的发展。PI在数学物理，分析，概率，高维几何以及这些领域之间的相互作用的研讨会，座谈会和国际会议上发表了许多演讲。

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.