Fluctuations, Resonances, and Critical Phenomena

本研究将探讨在无序状态下量子动力学的基本问题，以及与统计力学中的关键现象相关的主题。其中一个突出的挑战是开发数学工具来分析无序存在下的扩展状态和传导。特别令人感兴趣的是在相互作用的粒子系统中局域-离局域跃迁的可能性。这项工作将建立在PI最近取得的进展的基础上，该进展表明，在某些情况下，扩展状态通过在分离良好的定位中心之间的共振促进的隧道出现。在统计力学领域，将恢复对关键现象的研究，特别是在关键维度上的研究。目标是解决一些未解决的问题，特别是与phi（4d）场论有关的问题。关于这一主题的众所周知的非摄动结果尚未最终解决临界维数d = 4。研究还将旨在澄清一种新型过渡的可能存在，在这种过渡中，d = 2维系统中局部有序参数与淬火无序的耦合性质可能随着无序强度的变化而变化。PI过去在这个课题上的工作包括与J. Wehr共同证明了无序的舍入效应，以及最近与R. Greenberg和J. L. Lebowitz共同将其扩展到量子系统。预期的研究将结合概率（波动分析和大偏差理论）和分析（Hp函数空间和算子理论）的观点，以获得物理学感兴趣的结果。正如过去所证明的那样，揭示物理问题的挑战也丰富了数学领域。这项资助包括资助一名研究生和一名博士后，以及继续开展富有成效的合作。这些对拟议的研究和工作的培训部分都非常有价值。PI过去的工作促进了一些目前在相关领域活跃的研究人员的专业发展，本项目也将继续发挥这一作用。研究成果将通过出版物、讲座和研讨会的形式发表。PI协助组织了会议。

This research will address fundamental issues concerning quantum dynamics in the presence of disorder, and topics related to critical phenomena in statistical mechanics. One of the outstanding challenges is to develop mathematical tools for the analysis of the extended states and conduction in the presence of disorder. Of particular interest is the possibility of the localization - delocalization transition in systems of interacting particles. The work will build on the progress which was recently made by the PI which showed that in certain situations extended states emerge through tunneling facilitated by resonances between well separated localization centers. In the area of Statistical Mechanics, research will be resumed into critical phenomena, in particular at critical dimensions. The goal is to resolve some unsettled questions, in particular in relation to the phi(4d) field theory. The well known non-perturbative results on this topic did not yet address conclusively the critical dimension d = 4. Research will also be directed at clarifying the possible existence of a novel type of transition, at which the nature of the coupling of the local order parameter to the quenched disorder in d = 2 dimensional systems may change as the disorder strength is varied. The PI's past work on this subject included proof, jointly with J. Wehr, of the rounding effect of the disorder, and the recent extension of this to quantum systems, done jointly with R. Greenberg and J. L. Lebowitz. The envisioned research is expected to combine the perspectives of Probability (fluctuation analysis and large deviation theory) and Analysis (Hp function spaces and operator theory) for results which are of interest for Physics. As was demonstrated in the past, the challenge of shedding light on issues of physics enriches also fields of mathematics. This grant includes funds for a graduate student and a postdoc, and for the continuation of fruitful collaborations. These are highly valuable both for the proposed research and for the training component of the work. The PI's past work has contributed to the professional development of a number of men and women who are now active researchers in related fields, and this project will continue to serve that role as well. The research results will be presented through publications, lectures and workshops. The PI has assisted in the organization of meetings.