Localization Transitions in Effective Random Matrix Models

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This project aims to study the mathematics of quantum particles (such as electrons) in disordered media (such as doped semiconductors). The main point of focus is understanding how features like the geometry of the medium, the energy, and the strength of the disorder determine whether the material is conducting or not. For realistic models, these questions are far beyond the present mathematical state of the art so current research focuses on answering them for effective toy models.One important class of such toy models consists of matrices with random entries, whose variances decay away from the diagonal of the matrix. In this setting, the question reduces to studying the qualitative features of the eigenvectors of these random matrices. However, even for such simplified models, the current understanding is very far from complete. This project will study a subclass of these models with an imposed hierarchical structure that makes the analysis of conductance mathematically feasible. In particular, our recent works have shown that the localized (non-conducting) regime of the model can be completely understood using ideas from stochastic analysis. The principal goal of this project is to extend this work to the delocalized (conducting) regime and possibly even to more general models. Accomplishing this goal would yield a new addition to the extremely short list of toy models for which the localization transition can be proved rigorously.