Percolation Theory and Related Topics

渗滤理论及相关主题

• 批准号: 2246494
• 负责人: Thomas Hutchcroft
• 金额: $ 30万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246494&HistoricalAwards=false
• 关键词: Percolation; Theory; Related; Topics

Many large, complex systems arising in mathematics, physics, and elsewhere undergo phase transitions, where varying a parameter (e.g. the temperature or pressure) that describes the system at a small scale by a small amount through some special value causes an abrupt, qualitative change in the behaviour of the system on a large scale. Beyond the familiar examples of water freezing and boiling, phase transitions also occur in many other systems including ferromagnets, superconductors, superfluids, epidemics, and traffic. In each case, understanding when, how, and why the system undergoes a phase transition is of central importance in both theory and practice. Moreover, the basic mathematical principles underlying the occurrence of such phase transitions have much in common across these diverse situations, and the study of phase transitions has come to be recognised as a rich source of deep and beautiful pure mathematics that is of interest beyond and complementary to its practical origins. This project aims to develop our fundamental understanding of phase transitions and critical phenomena (the special properties exhibited by systems at the point of phase transition) through the study of probabilistic models. The project provides research training opportunities for graduate students. The project focuses on various probabilistic lattice models of statistical mechanics, including percolation, random walks, the Ising model, and the uniform spanning tree. In particular, the project aims to understand how the geometry of the underlying graph (e.g. the dimension of the lattice) influences the critical behaviour of the model, with focuses on quantitative aspects of critical phenomena and the comparison between short-range, long-range, and hierarchical models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在数学、物理和其他领域中出现的许多大型复杂系统都经历了相变，其中通过一些特殊值在小尺度上少量改变描述系统的参数（例如温度或压力）会导致系统在大尺度上的行为发生突然的质变。除了我们熟悉的水冻结和沸腾的例子，相变也发生在许多其他系统中，包括铁磁体、超导体、超流体、流行病和交通。在每种情况下，理解系统何时、如何以及为什么经历相变在理论和实践中都是至关重要的。此外，这种相变发生的基本数学原理在这些不同的情况下有很多共同点，并且相变的研究已经被认为是深刻而美丽的纯数学的丰富来源，它的兴趣超出了它的实际起源。本项目旨在通过概率模型的研究，发展我们对相变和临界现象（系统在相变点所表现出的特殊性质）的基本理解。本项目为研究生提供研究训练机会。该项目侧重于统计力学的各种概率格模型，包括渗透，随机漫步，伊辛模型和均匀生成树。具体而言，该项目旨在了解底层图形的几何形状（例如晶格的维度）如何影响模型的关键行为，重点关注关键现象的定量方面以及短程、远程和分层模型之间的比较。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。