Conformal Invariance and Two-Dimensional Statistical Physics

Gregory Lawler, University of Chicago

Abstract. A number of lattice models in two-dimensional statistical physics are conjectured to exhibit conformal invariance in the scaling limit at criticality. In this talk, I will try to explain what the previous sentence means, focusing on four elementary examples: simple random walk, self-avoiding walk, loop-erased random walk, and percolation. I will describe the limit-process Schramm-Loewner Evolution (SLE) and show how conformal invariance can be used to calculate quantities ("critical exponents") for the model. I will also describe why (in some sense) there is only a one-parameter family of conformally invariant limits. In conformal field theory, this family is parametrized by central charge. This talk is for a general mathematics audience; no knowledge of statistical physics is assumed.