Studies in Brownian Motion and Random Walk

布朗运动和随机游动的某些临界指数（交指数）与路径的几何和分形性质有密切的关系。 本研究员将研究在两个维度的交叉指数，特别是试图使用共形不变性来证明已使用非严格的共形场论来计算的指数的精确值。 这些思想也将应用于统计物理中临界现象中其他模型的连续极限。 我们的目标是给严格的参数使用共形不变的措施，从共形场论导出的路径上的结构。 研究者还将考虑随机游走的适当条件，以便其关键行为，即，它们的指数与布朗运动的指数相同。二维统计物理学中处于临界状态的许多系统，即，在或接近相变的温度下，被证明具有极限共形不变性的性质。 通过假设共形不变性，通常可以对描述系统定性行为的称为临界指数的某些量给出精确的预测。 无论是共形不变性假设还是假设共形不变性的指数预测都没有数学上的合理性。 研究人员将着眼于一个特定的系统，布朗运动路径的交叉点，并试图严格计算这些指数。 然后，他将这些想法扩展到更一般的系统，满足共形不变性。

9971220There is a close relationship between certain critical exponents, the intersection exponents, for Brownian motion and random walk and the geometric and fractal properties of the paths. This investigator will study the intersection exponent in two dimensions, in particular trying to use conformal invariance to prove exact values of the exponents that have been conjectured using nonrigorous conformal field theory. The ideas will also be applied to continuum limits of other models in critical phenomenon in statistical physics. The goal is to give rigorous arguments using conformally invariant measures on paths for conjectures derived from conformal field theory. The investigator will also consider appropriate conditions on random walks so that their critical behavior, i.e., their exponents, are the same as for Brownian motion.A number of systems in two dimensional statistical physics at criticality, i.e., at or near the temperature at which there is a phase transition, are conjectured to have the property of conformal invariance in the limit. By assuming conformal invariance it is often possible to give exact predictions for certain quantities called critical exponents which describe the qualitative behavior of the systems. Neither the conformal invariance assumption nor the predictions of the exponents assuming the conformal invariance has been mathematically justified. The investigator will look at a particular system, intersections of Brownian motion paths, and try to compute these exponents rigorously. He will then extend these ideas to more general systems that satisfy conformal invariance.