Abstract:

Critical points and singularities are encountered in the study of critical phenomena in probability and physics. We describe recent results (joint with Manolescu) concerning the values of critical points and the nature of the singularities for percolation and the random-cluster model. The main result is the universality of bond percolation on isoradial graphs. The key technique is the star-triangle transformation. The proofs are expected to extend to the random-cluster model on isoradial graphs.