Random Schrödinger operators arising in the studyof reinforced random processes

近年来，由于发现了与不同数学领域的联系，强化随机过程引起了科学界的极大兴趣，这些联系使得这些模型中的相变得到了严格的证明。在这个提议中，我们感兴趣的是最近发现的与某些随机Schrödinger算子的联系，这些算子用于模拟电子在无序介质中的传播。我们的目标是将发展良好的量子无序系统理论与这些新结果结合起来，并进一步利用这些联系来研究无序量子系统中的相变。我们希望我们的工作至少能提供Sabot和Zeng最近提出的一个问题的部分答案，并由此有助于理解无序固体理论中一个长期存在的开放问题，即无序量子系统的Anderson金属-绝缘体跃迁。

In recent years, reinforced random processes have attracted much interest from the scientific community due to the discovery of connections with different areas of mathematics that have allowed for rigorous proofs of phase transitions in these models. In this proposal, we are interested in a recently discovered connection with certain random Schrödinger operators, which are used to model the propagation of electrons in disordered media. Our goal is to combine the well developed theory of quantum disordered systems with these new results and exploit these connections further to study phase transitions in disordered quantum systems. We expect our work to provide at least a partial answer to a recent question formulated by Sabot and Zeng, and with this, to contribute to the understanding of a long-standing open problem in the theory of disordered solids, namely, the Anderson metal-insulator transition for disordered quantum systems.