RUI: Percolation Models

张 该项目集中在渗流，一个数学模型有应用到各种领域，如物理学，生物学，计算机科学和地质学。 渗滤过程通常取决于一个或多个参数。 物理性质可能会发生巨大变化，这取决于参数是高于还是低于临界值。 比如说， 假设我们把一个大的多孔固体浸入一桶水中。 显然，水如何渗透固体取决于孔隙的大小。 这种过程的一个简单的数学模型是 通过考虑孔隙分布在一些晶格中，并且以概率p或1-p打开或关闭来定义。对于行为突然改变的概率，存在临界阈值p（c）。 在阈值以下，水的渗透只是表面的，而在阈值以上，渗透是无限深的。 特别感兴趣的是临界阈值附近的行为。 最具挑战性的问题之一是证明无限穿透的概率表现为|p-p（c）|如果p接近p（c）。 这项研究将集中在数学上严格的渗流过程的精确解。 该项目集中在渗流，一个数学模型有应用到各种领域，如物理学，生物学，计算机科学和地质学。 渗滤过程通常取决于一个或多个参数。 物理性质可能会发生巨大变化，这取决于参数是高于还是低于临界值。 这项研究将集中在数学上严格的渗流过程的精确解。 研究中运用了概率论、图论、组合数学和函数分析等理论。***

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. ***