Random walks in irregular media

数学中对随机游走的研究至少可以追溯到1921年Polya的工作；他证明了一个简单的一维或二维随机漫步会无限次地回到起点，而在三维或多维中，它最终会离开它的初始邻域，永远不会回来。在大长度尺度上，随机游走可以用布朗运动来近似。扩散和偏微分方程（PDE）之间的联系是由于爱因斯坦；扩散过程的跃迁概率是热方程的解。数学家们在20世纪下半叶对这些普遍存在的联系进行了非常彻底的探索。***提议的研究是研究不规则介质中的随机游走（或其他扩散过程）。一个例子是无限道路网格，其中根据一些随机机制，一定比例的街道被阻塞。（在完全随机的情况下，这是渗透模型。）可能出现两种主要类型的行为。首先，障碍物（即不规则性）只具有局部效应，因此从很远的距离（即在大长度尺度上）来看，它们并没有发挥重要作用，随机步行者只是作为在均匀介质中移动的粒子出现。第二种可能性是，不规则性的影响在所有长度尺度上都持续存在。当极限对象具有分形特性时，这种情况经常发生。在这种情况下，介质的不规则性使粒子减速；它的位移比简单随机漫步的位移增长更慢。这种现象通常被称为“异常扩散”。拟议的研究将考虑这两种行为。在第一种情况下，问题是研究一般类型的不规则环境，并（只要它仍然是正确的）证明两种类型的大尺度（或更准确地说，限制）过程的规则。第一种规则是经典中心极限定理的扩展——粒子在大时间内的位移应该接近布朗运动。第二种规律性是粒子运动的跃迁密度可以由高斯分布的倍数来控制。如果这两种规律性都存在，那么我们就能很好地描述粒子的运动。在第二种情况下，总体目标是找到计算随机过程行为的方法，给定图形不规则性的知识。例如，人们想要计算“行走维度”，它描述了粒子从起点移动的平均速度。一个有趣的例子是一致生成树。这些问题在理论上是可以解决的，因为树的分支可以用随机行走路径来描述，而环路可以被删除。* * * * *

The study of random walks in mathematics goes back as least as far as the work of Polya in 1921; he showed that a simple random walk in one or two dimensions will return to its starting point infinitely often, while in 3 or more dimensions it will ultimately leave its initial neighbourhood, never to return. At large length scales random walks can be approximated by Brownian motion. The connection between diffusion and partial differential equations (PDE) is due to Einstein; the transition probabilities of the diffusion process are solutions to the heat equation. These connections, which hold in great generality, were explored very thoroughly in the second part of the 20th century by mathematicians. ***The proposed research is to study random walks (or other diffusion processes) in irregular media. An example would be an infinite road grid, where according to some stochastic mechanism a proportion of the streets are blocked. (In the case when this is completely at random this is the percolation model.) Two main types of behaviour can arise. The first is that the obstacles (i.e. irregularities) just have a local effect, so that viewed from a great distance (i.e. at large length scales) they play no essential role, and the random walker just appears as a particle moving in a homogeneous medium. The second possibility is that the effect of the irregularity persists at all length scales. This often occurs when the limiting object has fractal properties. In these circumstances the irregularity of the medium acts to slow down the particle; its displacement grows more slowly than that of the simple random walk. This phenomena is often called 'anomalous diffusion'.***The proposed research will consider both types of behaviour. In the first case, the problems are to study general kinds of irregular environments and (as far as it remains true) to prove two types of regularity for the large scale (or more precisely limiting) process. The first kind of regularity is an extension of the classical Central Limit Theorem - the particle displacement at large times should be close to a Brownian motion. The second kind of regularity is that the transition densities of the particle motion can be controlled by multiples of the Gaussian distribution. If both kinds of regularity occur then one has an very good description of the particle motion.***In the second case, the overall aim is to find methods for computing the behaviour of the random process given knowledge of the kind of graph irregularity. For exampe, one would like to compute the 'walk dimension', which describes the rate at which (on average) a particle moves from its starting point. One interesting set of examples are uniform spanning trees. These can be tackled theoretically because the branches of the trees can be described by random walk paths with the loops erased. *** **