Localization Transitions in Effective Random Matrix Models

该项目旨在研究无序介质（如掺杂半导体）中量子粒子（如电子）的数学。重点是了解介质的几何形状，能量和无序的强度等特征如何决定材料是否导电。对于现实模型，这些问题远远超出了目前的数学水平，因此目前的研究集中在回答有效的玩具模型。一类重要的玩具模型由随机元素的矩阵组成，其方差从矩阵的对角线开始衰减。在这种情况下，问题归结为研究这些随机矩阵的特征向量的定性特征。然而，即使是这样的简化模型，目前的理解是非常不完整的。这个项目将研究这些模型的一个子类，它具有一个强加的层次结构，使得电导的分析在数学上是可行的。特别是，我们最近的工作表明，局部（非导电）政权的模型可以完全理解使用随机分析的想法。该项目的主要目标是将这项工作扩展到离域（传导）机制，甚至可能扩展到更一般的模型。实现这一目标将产生一个新的除了极少数的玩具模型，其中本地化过渡可以证明严格。

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.