Mathematical Sciences: Studies in Random Walks

首席研究员将继续他在二维和三维随机行走路径结构方面的工作。感兴趣的特定对象是这些行走的临界指数：相交指数，测量两个独立行走相交的概率；化学指数，测量两点之间的最短子路径的长度；阻力指数；环擦除指数，测量从擦除环路得出的路径长度；以及与粒子在随机行走路径上扩散的行为有关的指数。研究人员还将研究由随机行走动力学形成的星团。首席研究人员将继续他在二维和三维随机行走路径的类分形结构方面的工作。这是概率论的一个领域，它在很大程度上借鉴了物理学和化学中的启发式模型来激励人们。这一领域的结果将影响概率界和理论物理界。一个感兴趣的对象是随机行走的相交指数，它衡量两个独立行走相交的概率。

The principal investigator will continue his work on the structure of random walk paths in two and three dimensions. The particular objects of interest are critical exponents for these walks: the intersection exponent, which measures the probability that two independent walks intersect; the chemical exponent, which measures the length of the shortest subpath between two points; the resistance exponent; the loop-erasing exponent, which measures the length of a path derived from erasing loop chronologically; and exponents related to the behavior of a particle diffusing on a random walk path. The investigator will also study clusters formed from random walk dynamics. The principal investigator will continue his work on the fractal-like structure of random walk paths in two and three dimensions. This is an area of probability theory that draws heavily on heuristic models in physics and chemistry for motivation. Results in this area will impact both the probability community and the theoretical physics community. One object of interest is the intersection exponent of a random walk, which measures the probability that two independent walks intersect.