Critical Phenomena and Stochastic Geometry

拟议的研究涉及分析在具有有趣物理学的模型中发挥重要作用的问题，特别是凝聚态领域的问题。 我们的目标是促进对所涉及的物理学的理解，并开发具有相关性和内在兴趣的数学结构。特别注意的是：i）随机几何的激发相关的系统，这是至关重要的意义上的统计力学，和ii）的局域化无序的存在下量子算符的去局域化过渡。分形结构在临界系统中的关联涨落和低温下量子自旋链中的关联性质中起着重要作用。 建议继续发展基本的工具来处理涌现的随机几何结构，研究从渗流到经典和量子自旋模型的系统中的一些模型特定的和维度相关的性质，并阐明与共形场论的关系。扩展激发在传导现象中起着基础性的作用。 所提出的工作的另一个目标是分析去本地化过渡，即，在广泛无序的系统中形成扩展状态。

The proposed research concerns the analysis of issues which play important roles in models with interesting physics, particularly of the area of condensed matter. The goal is to contribute to the understanding of the physics involved, and also to develop mathematical structures which are of relevance as well as of intrinsic interest. Particular attention will be given to: i) the stochastic geometry of excitations related to the correlations in systems which are critical in the sense of statistical mechanics, and ii) the de-localization transition for quantum operators in the presence of localizing disorder. Fractal structures play important roles in correlated fluctuations in critical systems, and in the nature of correlations in quantum spin chains at low temperatures. It is proposed to continue to develop the basic tools for dealing with the emergent stochastic geometric structures, to study some model-specific and dimension-dependent properties in systems ranging from percolation to classical and quantum spin models, and to elucidate the relations with conformal field theory. Extended excitations play fundamental role in conduction phenomena. The other goal of the proposed work is to analyze the de-localization transition, i.e., the formation of extended states in systems with extensive disorder.