Abstract

Critical loop O(n) models are conjectured to be conformally invariant in the scaling limit. In this talk, we focus on connection probabilities for loop O(n) models in polygons. Such probabilities can be predicted using two families of solutions to chordal Belavin-Polyakov-Zamolodchikov(BPZ) equations: Coulomb gas integrals and SLE pure partition functions. The conjecture is proved to be true for the critical Ising model, FK-Ising model, percolation, and uniform spanning tree.  Recent progress of radial BPZ equations will also be discussed.