Consider the standard Ising model on a finite n x n grid at low temperature. If the boundary spins are held fixed equal to +1 it is believed that the mixing time of the corresponding Glauber dynamics (Gibbs sampler) is poly(n). Although such a result is still far from being proved, recently there has been some exciting progress using the censoring inequality by Peres and P. Winkler together with inductive schemes. The final outcome is a
quasi-poly(n) bound valid for all temperatures below the critical one.
Based on joint work with F.L.Toninelli, and F.L.Toninelli, E. Lubetzky and A. Sly.