RUI: Percolative models

渗流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。它们也被大量用于物理学、生物学和计算机科学。在这一建议中，张重点介绍了三个基本的渗透模型：定向渗透模型，第一通道渗透模型和渗透模型。该研究使用概率定理（例如，鞅概率定理、马尔可夫链、CLT定理和相关不等式）;图形定理（例如，对于对偶性、晶格表面和平面性）;组合学（例如，分格，重整化）;以及泛函分析（例如，该项目将为张的本科生在工程、计算机科学和数学领域提供教学经验。由于渗透理论中的问题和开放性问题对学生来说很容易理解，因此建议的项目也为学生提供了一个对渗透问题进行研究的机会。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，学生们会发现在这一重要领域进行调查很有吸引力。总之，本项目概述的学生研究活动将大大有助于提高该机构的教育质量。拟议的项目还提高了对技术和科学的认识。由于渗流理论涉及数值计算和模拟，该项目需要计算科学的新技术。

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.