Mathematical Sciences: Percolative Models

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9618128 Zhang The project concentrates on percolation theory, a mathematical theory used to describe transitions of physical systems. Percolation theory has a variety of applications to solid physics, biology, computer science, and geology. A percolation process typically depends on one or more parameters. A dramatic change in physical properties may occur as a critical parameter value is passed. The research will focus on the behavior of percolative models in the following three areas: percolation near, above and below the critical thresholds. In particular, the project will investigate mathematically rigorous exact solutions for the percolative models. The research makes use of probability theory (the moment estimations, the ergodic theory, correlation and martingale inequalities and stochastic ordering), graph theory (duality, the fractal dimension), combinatorics (partition lattices, distributive lattices), and function analysis (the real analyticity). The project will use these mathematical tools to advance our rigorous understanding of critical phenomena. The project concentrates on percolation, a mathematical model used to describe transitions of physical systems. Percolation theory has a variety of applications to solid physics, biology, computer science, and geology. A percolation process typically depends on one or more parameters. A dramatic change in physical properties may occur as a critical parameter value is passed. For example, suppose we immerse a large porous solid in a bucket water. Clearly, how water penetrates the solid depends on the size of the pores of the solid. A simple mathematical model of such a process is defined by taking the pores to be distributed in some regular manner, and to be open or closed with probabilities p or 1-p. There is a critical threshold, for the probability p at which the behavior changes abruptly. For values of p below the critical value the water penetration is only superficial and above it the penetration is arbitrarily deep. The specific behavior near the critical threshold is more complicated. One of the most challenging problems is to give a mathematical description of deep penetration near the critical threshold. The research will focus on three areas: the behaviors of percolation near the critical threshold, above the critical threshold and below the critical threshold. In particular, the project will investigate mathematically rigorous exact solutions for the percolation process. The research makes use of probability theory, graph theory, combinatorics and function analysis. The project will use these mathematical tools to advance our rigorous understanding of critical phenomena.