Spectral Properties of Random Band Matrices and Related Questions

该研究项目涉及数学和物理相关领域，包括矩阵理论、概率论、经典分析和算符理论，以及量子场论和超对称等物理主题。超对称方法在理论物理特别是粒子物理中有着广泛的应用，但其严密的数学基础仍然是数学家们所面临的挑战。该项目的目的是发展对超对称的广泛的数学理解。更准确地说，随机频带矩阵（RBM）表示d维晶格上的量子系统，其随机量子跃迁振幅有效到某个W阶的距离，这被称为带宽。它们是研究无序系统的特征值统计和量子传播的自然中间模型；它们在维格纳随机矩阵和随机薛定谔算子之间进行插值。Wigner矩阵系综表示没有空间结构的模型，其中任意两个点之间的量子跃迁速率是独立的同分布随机变量。相比之下，随机薛定谔算子除了在d维晶格中的大盒子上的确定性拉普拉斯算子外，只有一个随机对角势。RBM的主要特点是可以在三维或多维度上对Anderson金属-绝缘体相变进行建模。我们推测，通过改变带宽W，即使在一维上也可以研究RBM的交叉；然而，在严格的数学层面上，这个问题仍然是开放的。研究者将利用超对称方法结合强大的分析和统计力学技术，包括鞍点分析、传递算子和多尺度分析来解决这个问题。

This research project intertwines mathematical and physics-related areas including matrix theory, probability, classical analysis and operator theory, as well as topics in physics such as quantum field theory and supersymmetry. The method of supersymmetry is widely used in theoretical physics, especially in particle physics, but its rigorous mathematical foundation is still a challenge for mathematicians. The aim of the project is to develop a broad mathematical understanding of supersymmetry.More precisely, random band matrices (RBM) represent quantum systems on a d-dimensional lattice with random quantum transition amplitudes effective up to distances of some order W, which is called a bandwidth. They are natural intermediate models in the study eigenvalue statistics and quantum propagation of disordered systems; they interpolate between Wigner random matrices and random Schrodinger operators. Wigner matrix ensembles represent models without spatial structure, where the quantum transition rates between any two sites are independent identically distributed random variables. In contrast, random Schrodinger operators have only a random diagonal potential in addition to the deterministic Laplacian on a large box in a d-dimensional lattice. The main feature of RBM is that they can be used to model the Anderson metal-insulator phase transition in three or more dimensions. It is conjectured that the crossover for RBM can be investigated even in dimension one by varying the bandwidth W; however, on a mathematical level of rigor the question is still open. The investigator is going to attack the problem using the supersymmetry approach in combination with powerful analytic and statistical mechanics techniques, including saddle point analysis, transfer operators, and multi-scale analysis.