Mathematical Sciences: Percolative Models

小行星9618128 该项目集中在渗流理论，一个数学理论，用于 描述物理系统的转换。渗流理论有多种应用， 固体物理学、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剧烈变化 因为传递了关键参数值。这项研究将集中在行为的 在以下三个方面的渗透模型：附近，上方和下方 临界阈值。特别是，该项目将研究数学上严格的精确 解决方案的模型。该研究利用概率论（ 矩估计，遍历理论，相关和鞅不等式， 随机排序），图论（对偶，分形维数），组合学（划分 格、分配格）和函数分析（真实的解析性）。该项目将 用这些数学工具来推进我们对关键的 现象。 该项目集中在渗流，一个数学模型，用于描述 物理系统的转变。渗流理论在固体中有许多应用 物理、生物、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。物理性质的急剧变化可能会发生， 关键参数值被传递。例如，假设我们将一个大的多孔固体 在一桶水里。显然，水如何渗透固体取决于孔隙的大小 的固体。这样一个过程的一个简单的数学模型是通过将孔隙 以某种规则的方式分布，以概率p或 1-p。对于行为发生变化的概率p，存在一个临界阈值 突然当p值低于临界值时，水的渗透只是表面现象 而在其上的穿透深度是任意的。临界点附近的特定行为 门槛比较复杂。最具挑战性的问题之一是给出一个 临界阈值附近深穿透的数学描述。这项研究将 重点研究了三个方面：临界阈值附近的渗流行为，临界阈值以上的渗流行为， 临界阈值和临界阈值以下。特别是，该项目将调查 数学上严格的渗流过程的精确解。这项研究使 运用概率论、图论、组合学和函数分析。项目 将使用这些数学工具来推进我们对关键的 现象。

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.