RUI: Percolation Model

0405150Zhang 渗透模型形成了概率论的一个非常活跃的子领域。他们从物理学、生物学和计算机科学中汲取了很多灵感和动力。它们还广泛应用于物理学、生物学、工业和计算机科学领域。该项目重点关注以下渗滤问题： 一般渗滤模型：(a) 幂律和标度关系。 (b) 初始无限簇的中心极限定理。 (c) 对任意单词的渗透。 (d) 关于定向渗透中右边缘的无限可微性。第一段渗透：（a）时间常数的不可微性。 (b) 偏差较大。该研究利用了概率定理（鞅、马尔可夫链、CLT 定理、相关不等式）、图定理（对偶性、晶格曲面、平面性）、组合数学（分格、重正化）和泛函分析（复变量）。 该项目对研究、教育和传播产生广泛影响，通过以下方式促进科学和技术理解：首先，渗滤理论是数学中最重要的学科之一。事实上，渗流理论中的问题和开放性问题很容易陈述，但其解决显然很困难，需要创新的方法。此外，渗滤理论还可以直接应用于生物学、工业和计算机科学。张提出了该领域的六个传统问题，并提出了解决这些问题的新方法。其次，该提案提升了张教授本科生在工程、计算机科学和数学领域的教学经验。它还为学生提供了进行渗滤问题研究活动的机会。此外，该提案还增强了科学技术理解。由于渗流理论涉及数值计算和模拟，因此需要计算科学的技术进步。

0405150Zhang Percolative models form a very active subfield of probability theory. They have drawn much of their inspiration and motivation from physics, biology and computer science. They have also been used heavily in physics, biology, industry, and computer science. This project focuses on the following percolation questions: General percolation model: (a) The power laws and scaling relations. (b) The central limit theorems for the incipient infinite cluster. (c) Percolation on arbitrary words. (d) On the infinite differentiability of the right edge in oriented percolation. First passage percolation: (a) Non-differentiability of the time constant. (b) The large deviations. The research makes use of probability theorems (martingale, Markov chains, CLT theorem, correlation inequalities), graph theorems (duality, lattice surfaces, planarity), combinatorics (partition lattice, renormalization) and functional analysis (complex variables). The project has a broad impact on research, education and dissemination to advance scientific and technological understanding in the following ways: First, percolation theory is one of most important subjects in mathematics. In fact, issues and open questions in percolation theory are easy to state, but whose solutions are apparently difficult and require innovative methods. Furthermore, percolation theory is also directly applicable in biology, industry, and computer science. Zhang presents six traditional problems in this field and proposed new methods to solve them. Second, the proposal promotes teaching experiences for Zhang's undergraduate students in areas of engineering, computer science and mathematics. It also offers an opportunity for the students to conduct research activities on percolation problems. Moreover, the proposal also enhances scientific and technological understanding. Since percolation theory involves numerical computations and simulations, technological advances in computing science are required.