RUI: Percolation Models

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Zhang The project concentrates on percolation, a mathematical model has applications to a variety of areas, such as physics, biology, computer science, and geology. A percolation process typically depends on one or more parameters. There may be a dramatic change in physical properties depending on whether a parameter is above or below a critical value. For example, suppose we immerse a large porous solid in a bucket of water. Clearly, how water penetrates the solid depends on the size of the pores. A simple mathematical model of such a process is defined by considering the pores to be distributed in some lattice, and to be open or closed with probabilities p or 1-p. There is a critical threshold, p(c), for the probability at which the behavior changes abruptly. Below the threshold the water penetration is only superficial and above it the penetration is infinitely deep. Of special interest is behavior near the critical threshold. One of the most challenging problems is to show that the probability of infinite penetration behaves like a power of |p-p(c)| if p is near p(c). This research will focus on mathematically rigorous exact solutions for the percolation process. The project concentrates on percolation, a mathematical model has applications to a variety of areas, such as physics, biology, computer science, and geology. A percolation process typically depends on one or more parameters. There may be a dramatic change in physical properties depending on whether a parameter is above or below a critical value. This research will focus on mathematically rigorous exact solutions for the percolation process. The research makes use of probability theory, graph theory, combinatorics and function analysis. ***