Percolation Theory and Related Topics

在数学、物理和其他领域中出现的许多大型、复杂的系统都会经历相变，在这种情况下，通过某个特定值将描述系统的参数(例如温度或压力)在小范围内小幅改变会导致系统在大范围内的行为发生突然的、质的变化。除了熟悉的水冻结和沸腾的例子外，相变还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。在每一种情况下，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之处，相变的研究已经被认为是深奥而美丽的纯数学的丰富来源，这是对其实际起源之外的兴趣和补充。这个项目的目的是通过对概率模型的研究，加深我们对相变和临界现象(系统在相变点表现出的特殊性质)的基本理解。该项目为研究生提供了研究培训机会。该项目专注于统计力学的各种概率晶格模型，包括渗流、随机游动、伊辛模型和均匀生成树。特别是，该项目旨在了解基本图形的几何形状(例如网格的尺寸)如何影响模型的临界行为，重点放在临界现象的量化方面以及短期、远程和分层模型之间的比较。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.