Critical Phenomena and Stochastic Geometry

提出的研究涉及的问题，在具有有趣的物理模型中发挥重要作用的分析，特别是在凝聚态领域。目标是促进对所涉及的物理学的理解，并发展相关的数学结构以及内在的兴趣。将特别注意：i)与统计力学意义上至关重要的系统中相关的激励的随机几何，以及ii)局域无序存在下量子算子的去局域跃迁。分形结构在临界系统的相关涨落和低温下量子自旋链的相关性质中起着重要作用。建议继续发展处理紧急随机几何结构的基本工具，研究从渗透到经典和量子自旋模型的系统中的一些特定模型和维度相关的性质，并阐明与共形场理论的关系。扩展激励在传导现象中起着重要的作用。提出的工作的另一个目标是分析去局域化转变，即广泛无序系统中扩展状态的形成。

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.