RUI: Percolative models

渗透模型是概率论中非常活跃的子领域。他们从物理学、生物学和计算机科学中获得了很多灵感和动力。它们在物理学、生物学和计算机科学中也被大量使用。在这篇论文中，Zhang重点研究了三种基本的渗流模型：定向渗流模型、第一通道渗流模型和渗流模型。研究使用概率定理（如鞅概率定理、马尔可夫链、CLT定理和相关不等式）；图定理（如对偶、格面和平面）；组合学（如：分格、重整化）；和功能分析（例如，复杂变量）。该项目旨在促进张教授的本科生在工程、计算机科学和数学领域的教学经验。由于渗透理论中的问题和悬而未决的问题对学生来说很容易理解，因此拟议的项目也为学生提供了对渗透问题进行研究活动的机会。此外，渗透理论也直接适用于生物、工业和计算机科学。因此，学生们会发现在这一重要领域进行调查是很有吸引力的。总之，本项目概述的学生研究活动将对该机构的教育质量做出重大贡献。拟议的项目还加强了对技术和科学的了解。由于渗流理论涉及数值计算和模拟，该项目需要新的计算科学技术。

Percolative models are very active subfields of probability theory. They have drawn much of their inspiration and motivation from physics, biology, and computer science. They also have been used heavily in physics, biology, and computer science. In this proposal, Zhang focuses on three basic percolative models: an oriented percolation model, a first passage percolation model, and a percolation model. The research uses probability theorems (e.g., the martingale probability theorem, Markov chains, the CLT theorem, and correlation inequalities); graph theorems (e.g., for duality, lattice surfaces, and planarity); combinatorics (e.g., partition lattice, renormalization); and functional analysis (e.g., complex variables).The proposed project promotes teaching experiences for Zhang's undergraduate students in the areas of engineering, computer science, and mathematics. Because issues and open questions in percolation theory are easy for students to understand, the proposed project also offers students an opportunity to conduct research activities on percolation problems. Furthermore, percolation theory is also directly applicable in biology, industry, and computer science. Thus, students will find it appealing to pursue investigations in this important area. In conclusion, the activities of students' research outlined in this project would contribute significantly to the quality of education at this institution. The proposed project also enhances technological and scientific understanding. Because percolation theory involves numerical computations and simulations, the project requires new technology in computing science.