Spectral Properties of Random Band Matrices and Related Questions

Princeton University, Princeton NJ

Investigators

Abstract

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.