From Random Walks to Critical Phenomena and Conformal Field Theory

In class, we approached critical phenomena from the view point of correlation functions and discussed renormalization group methods for obtaining values of critical exponents. Here I discuss a parallel idea that studies critical phenomena from properties of random curves which form the domain walls of the critical system. It will be shown that such random curves obey a stochastic differential equation which can be regarded as a conformal mapping. By studying the behavior of these conformal maps, such as the likelihood that a random walk connects two points or encompasses a certain point, one can make connections with quantities in conformal field theory. In addition, it will be shown that the critical exponent can be regarded as the diffusion constant for such random walks. This is a new field which has recently attracted the attention of theoretical physicists and many questions still remain. A general overview is presented here.