Studies in Brownian Motion and Random Walk

9971220布朗运动和随机行走的某些临界指数，即交点指数，与路径的几何性质和分形性质有密切的关系。这个研究者将研究二维的交点指数，特别是试图用共形不变性来证明用非严格共形场论推测的指数的精确值。这些思想也将应用于统计物理中其他临界现象模型的连续体极限。我们的目标是用共形不变测度给出从共形场论推导出的猜想路径上的严格论证。研究者还将考虑随机漫步的适当条件，使它们的临界行为，即它们的指数，与布朗运动相同。在二维统计物理中，在临界温度下，即在有相变的温度或温度附近，推测许多系统在极限处具有保形不变性。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。保形不变的假设和指数在保形不变前提下的预测都没有得到数学上的证明。研究者将着眼于一个特定的系统，布朗运动路径的交叉点，并尝试严格地计算这些指数。然后他将这些思想扩展到满足共形不变性的更一般的系统。

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.