Critical Manifolds for Percolation and Potts Models from Graph Polynomials

Jesper Jacobsen

The first parameter to be fixed when studying a phase transition is the critical temperature. Somewhat surprisingly, this parameter is only known analytically for the simplest two-dimensional models (Ising model), or for more complicated models (Potts and O(n) vector models) on the simplest possible lattices. The known critical temperatures are invariably given by simple algebraic curves. Some of these results have very recently been proved mathematically by the technique of discrete holomorphicity.

http://scgp.stonybrook.edu/video_portal/video.php?id=325

Simons- Program: Conformal Geometry