Critical Phenomena and Stochastic Geometry

这项拟议的研究涉及对具有有趣物理的模型中扮演重要角色的问题的分析，特别是对凝聚态物质领域的分析。目标是促进对所涉及的物理学的理解，并发展具有相关性和内在趣味性的数学结构。将特别注意：i)与系统中的关联有关的激发的随机几何，这在统计力学意义上是关键的，以及ii)在局域无序存在的情况下量子算符的非定域跃迁。在临界系统中的关联涨落中，以及在低温下量子自旋链的关联性质中，分形结构起着重要的作用。建议继续发展处理涌现的随机几何结构的基本工具，研究从渗流到经典和量子自旋模型的一些特定于模型的和依赖于维度的性质，并阐明与共形场论的关系。扩展激发在传导现象中起着重要的作用。这项工作的另一个目标是分析非定域化转变，即具有广泛无序的系统中扩展态的形成。

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.