Fluctuations, Resonances, and Critical Phenomena

点击翻译按钮获取中文摘要

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.