RUI: Percolation Model

点击翻译按钮获取中文摘要

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.