Percolative Models

0071635张本项目集中于渗流理论，一种用于描述物理系统转变的数学理论。 逾渗理论在固体物理学、生物学、计算机科学和地质学中有着广泛的应用。 逾渗过程通常取决于一个或多个参数，并且当通过临界参数值时，可能发生物理性质的显著变化。 本研究主要从以下三个方面来研究扩散模型的行为：渗流模型、首次通过渗流模型和扩散过程。 更确切地说，研究利用概率论（矩估计，遍历理论，相关和鞅不等式和随机序），CLT定理，图论（对偶，分形维数），组合学（划分格，分配格），和功能分析（真实的解析性）。 该项目将使用这些数学工具来推进对临界现象的严格理解。该项目专注于渗透，一种用于描述物理系统转变的数学模型。 逾渗理论在固体物理学、生物学、计算机科学和地质学中有着广泛的应用。 逾渗过程通常取决于一个或多个参数，并且当通过临界参数值时，可能发生物理性质的显著变化。 例如，假设我们将一个大的多孔固体浸入水桶中。 显然，水如何渗透固体取决于固体孔隙的大小。 这种过程的一个简单的数学模型是这样定义的：取孔隙以某种规则的方式分布，并以概率p或1-p打开或关闭。对于行为突然改变的概率，存在一个临界阈值，低于该阈值，水渗透只是表面的，高于该阈值，水渗透是任意深度的。 临界阈值附近的行为更为复杂。最具挑战性的问题之一是给出临界阈值附近深穿透的数学描述。 本研究将集中在三个方面：渗流模型，第一通道渗流模型，和分解过程。 特别是，该项目将研究渗透过程的数学严格的精确解。 研究中运用了概率论、图论、组合数学和泛函分析等理论。 该项目将使用这些数学工具来推进对关键现象的严格理解。

0071635ZhangThis 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.