Mathematical Sciences: Studies in Random Walk

劳勒将调查一些问题， 简单随机游动与调和测度的几何性质。 研究中包括对交叉点的分析 指数和电阻指数，类似于 数学物理中的临界指数 扩散限制 聚合是基于随机游走动力学的聚类模型。 主要目标是证明关于增长率的定理， 形状和这些集群的收敛速度。 随机游走模型被用于物理学的许多领域 和其他自然科学。 研究者将继续 研究，以严格理解的数学性质， 其中一些模型。 特别令人感兴趣的是 随机游动和带约束游动的交集 出现在统计力学和临界现象中。 另一 区域是集群形成，特别是使用 随机游走动力学 这些模型是在研究 枝晶生长和界面的研究。

Lawler will investigate a number of questions about the geometric properties of simple random walk and harmonic measure. Included in the research is an analysis of the intersection exponent and resistance exponents, which are analogous to critical exponents in mathematical physics. Diffusion limited aggregation is a cluster model based on random walk dynamics. The main goal will be to prove theorems about the growth rate, shape, and convergence rate for these clusters. Random walk models are used in a number of areas of physics and other natural sciences. The investigator will continue research to understand rigorously the mathematical properties of some of these models. Of particular interest are problems on the intersection of random walks and walks with constraints which arise in statistical mechanics and critical phenomenon. Another area is cluster formation, especially clusters which grow using random walk dynamics. These models arise in the study of dendritic growth and in the study of interfaces.