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.
