Percolative Models

University Of Colorado At Colorado Springs, Colorado Springs CO

Investigators

Abstract

0071635 Zhang This 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, and a dramatic change in physical properties may occur as a critical parameter value is passed. This research will focus on the behaviors of percolative models in the following three areas: percolation model, first passage percolation model, and the percolative process. More precisely, the research makes use of probability theory (the moment estimations, the ergodic theory, correlation and martingale inequalities and stochastic ordering), the CLT theorem, graph theory (duality, the fractal dimension), combinatorics (partition lattices, distributive lattices), and functional analysis (the real analyticity). The project will use these mathematical tools to advance in a rigorous understanding of the critical phenomena. This 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, and 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 probability at which the behavior changes abruptly, below which the water penetration is only superficial and above which it is arbitrarily deep. The 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. This research will focus on three areas: percolation model, first passage percolation model, and percolative process. 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 functional analysis. The project will use these mathematical tools to advance in a rigorous understanding of the critical phenomena.