Random motion in random or irregular media

这项工作的目的是研究随机运动的粒子在不规则介质中的行为。这可以看作是颗粒在岩石或土壤中扩散的一个简单模型。对这类模型进行严格的数学分析已被证明是困难的，随着新技术的开发和应用，最近出现了许多活动。其目的是了解介质中的不规则性如何影响扩散的性质，特别是估计这样的量，如一个粒子在某一点开始到达另一粒子在某一时刻的概率。渗流模型是随机介质的简单数学模型。对于正方形格子的每条边，我们抛出一枚硬币，并以概率p宣布这条边是开放的，否则它是闭合的。随着p的增加，疏散星团的大小也会增加，直到达到一个特定值时才会形成无限大的星团。这个无限团簇具有分形结构，这使得在这个团簇中的扩散与在正规空间中的扩散有很大的不同，我们希望在这个模型和相关模型中对这个扩散有进一步的了解。

The aim of this work is to study the behaviour of a randomly moving particle in an irregular medium. This can be regarded as a simple model for diffusion of particulates in rock or soil. The mathematically rigorous analysis of such models has proved difficult and there has been a lot recent activity as new techniques have been developed and applied. The aim is to develop an understanding of how the irregularities in the medium affect the properties of the diffusion and in particular to estimate such quantities as the probability that a particle started at some point has reached another at a certain time. The percolation model is a simple mathematical model for a random medium. For each edge of the square lattice we toss a coin and declare the edge to be open with a probability p, otherwise it is closed. As p increases the size of the open clusters increases until at a specific value an infinite cluster will form. This infinite cluster has fractal structure which makes diffusion in this cluster rather different from that in normal space and we hope to gain a further understanding of this diffusion in this and related models.