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.

数学，物理和其他地方出现的许多大型复杂系统经历了相变，其中改变了一个参数（例如温度或压力），该参数以少量数量描述了系统，从而导致某些特殊值会导致系统行为的突然变化。 除了熟悉的水冷冻和沸腾的例子外，相位过渡还发生在许多其他系统中，包括铁磁铁，超导体，超流体，超流动，流行病和交通。在每种情况下，了解系统在理论和实践中都至关重要。此外，在这些不同的情况下，发生这种相变发生的基本数学原理具有很大的共同点，并且对相转换的研究已被公认为是丰富而美丽的纯粹数学的丰富来源，这是超越其实际起源的互补的。该项目旨在通过研究概率模型的研究来发展我们对相变和关键现象的基本理解（系统在相变时表现出的特殊特性）。该项目为研究生提供了研究培训机会。该项目着重于统计力学的各种概率晶格模型，包括渗透，随机步行，ISING模型和统一跨越树。特别是，该项目旨在了解基础图的几何形状（例如，晶格的维度）如何影响模型的批判行为，重点关注批判现象的定量方面以及短期，远程，远程和等级模型之间的比较。这些奖项通过评估NSF的合法任务和构建范围，这是NSF的范围的影响力，是依据的依据。 标准。