Fluctuations, Resonances, and Critical Phenomena

这项研究将解决在无序存在的情况下有关量子动力学的基本问题，以及与统计力学中的临界现象有关的主题。其中一个突出的挑战是开发数学工具来分析无序存在时的扩展状态和传导。特别令人感兴趣的是相互作用粒子系统中局域-离域转变的可能性。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，通过良好分离的局域中心之间的共振促进的隧道效应，出现了延伸态。在统计力学领域，将恢复对关键现象的研究，特别是关键方面的研究。目标是解决一些悬而未决的问题，特别是与Phi(4d)场理论有关的问题。关于这一主题的众所周知的非微扰结果还没有最终解决临界维度d=4。研究还将致力于澄清可能存在的一种新类型的跃迁，在这种类型的跃迁下，d=2维系统中局域有序参数与猝灭无序的耦合性质可能随着无序强度的变化而改变。PI过去在这个问题上的工作包括与J.Wehr共同证明无序的舍入效应，以及最近与R.Greenberg和J.L.Lebowitz共同完成的将其扩展到量子系统的证明。预期的研究将结合概率(涨落分析和大偏差理论)和分析(HP函数空间和算子论)的角度，得到物理学感兴趣的结果。正如过去所证明的那样，阐明物理问题的挑战也丰富了数学领域。这笔赠款包括资助一名研究生和一名博士后，以及继续进行卓有成效的合作。这些对于拟议的研究和工作的培训部分都非常有价值。国际和平研究所过去的工作促进了一些男性和女性的专业发展，他们现在是相关领域的活跃研究人员，该项目也将继续发挥这一作用。研究成果将通过出版物、讲座和研讨会进行介绍。国际和平协会协助组织了会议。

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.