The aim of this project is to study random Schrödinger operators that have arisen in recent years in connection with random walks in different settings, namely, random walks in random environments, random quantum walks, and random walks in highly connected networks. These settings all have in common the central role of random operators. We aim to exploit the extensively developed theory of random Schrödinger operators to study these models, and obtain a complete description of their spectral and dynamical properties. This will require the use and development of tools at the interface of probability, analysis and physics. We expect that this will provide information for phase transitions in the aforementioned models.

Keywords: Random operators, Anderson localization, disordered quantum systems, Integrated Density of States, random quantum walks, reinforced random processes, fractional random operators, multiscale analysis, spectral theory