RUI: Percolation Model

渗透模型是概率论中一个非常活跃的分支。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。它们也被大量应用于物理学、生物学、工业和计算机科学。本项目主要研究以下渗流问题：一般渗流模型：(a)幂律和标度关系。(b)初始无限簇的中心极限定理。(c)对任意词语的渗透。(d)定向渗流中右边的无穷可微性。第一段渗流：(a)时间常数不可微。(b)大偏差。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、格面、平面性）、组合学（分格、重整化）和泛函分析（复变量）。该项目在研究、教育和传播方面具有广泛的影响，可以通过以下方式促进科学和技术的认识：首先，渗透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表述，但解决这些问题显然很困难，需要创新的方法。此外，渗透理论也直接适用于生物、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，该提案促进了张的本科生在工程、计算机科学和数学领域的教学经验。它也为学生提供了一个进行渗透问题研究活动的机会。此外，该建议还增强了科学和技术的认识。由于渗流理论涉及数值计算和模拟，因此需要计算机科学的技术进步。

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.