Percolative Models

[0071635]张本项目主要研究渗透理论，这是一种用于描述物理系统过渡的数学理论。渗流理论在固体物理学、生物学、计算机科学和地质学中有着广泛的应用。渗透过程通常取决于一个或多个参数，当通过一个关键参数值时，物理性质可能会发生巨大变化。本文将从渗流模型、第一通道渗流模型和渗流过程三个方面对渗流模型的行为进行研究。更确切地说，该研究利用了概率论（矩估计、遍历理论、相关和鞅不等式以及随机排序）、CLT定理、图论（对偶性、分形维数）、组合学（划分格、分配格）和泛函分析（实解析性）。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。这个项目专注于渗透，一个用于描述物理系统过渡的数学模型。渗流理论在固体物理学、生物学、计算机科学和地质学中有着广泛的应用。渗透过程通常取决于一个或多个参数，当通过一个关键参数值时，物理性质可能会发生巨大变化。例如，假设我们将一个大的多孔固体浸入一桶水中。显然，水渗透固体的方式取决于固体孔隙的大小。这一过程的一个简单的数学模型是这样定义的：将孔隙以某种规则的方式分布，并以p或1-p的概率打开或关闭。有一个临界阈值，表示行为突然发生变化的概率，低于此阈值水的渗透只是表面的，高于此阈值水的渗透是任意深的。在临界阈值附近的行为更为复杂。其中最具挑战性的问题之一是在临界阈值附近给出深穿透的数学描述。本文的研究主要集中在三个方面：渗流模型、第一通道渗流模型和渗流过程。特别是，该项目将研究数学上严格的精确解决方案的渗透过程。本研究运用了概率论、图论、组合学和泛函分析。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.