RUI: Percolative models

渗流模型是概率论中非常活跃的一个分支。他们的灵感和动力主要来自物理学、生物学和计算机科学。它们还被广泛应用于物理学、生物学和计算机科学。在这一提议中，张着重于三个基本的渗流模型：定向渗流模型、首次通过渗流模型和渗流模型。这项研究使用了概率理论(例如，鞅概率定理、马尔可夫链、CLT定理和相关不等式)、图论(例如，对偶性、格子曲面和平面性)、组合学(例如，分割格、重整化)和泛函分析(例如，复变量)。拟议的项目促进了张的本科学生在工程、计算机科学和数学领域的教学经验。由于渗流理论中的问题和开放问题对学生来说很容易理解，建议的项目也为学生提供了一个开展渗流问题研究活动的机会。此外，渗流理论也直接适用于生物学、工业和计算机科学。因此，学生们将发现在这一重要领域进行调查是有吸引力的。总而言之，该项目中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理解。由于渗流理论涉及数值计算和模拟，该项目需要计算科学方面的新技术。

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.