Crossing Probabilities, Modular Forms and Anchored Percolation Clusters

Peter Kleban, University of Maine


We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic L÷wner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of 'higher-order modular form' arises and its properties are discussed briefly. We will also describe some more recent work on clusters in the half-plane connecting three points.