Glauber dynamics for the Ising Model on the Complete Graph

We study the Glauber dynamics for the Ising model on the complete graph on n vertices, also known as the Curie-Weiss model. For the high-temperature regime (\beta < 1), we prove that the dynamics exhibits a cutoff: the total variation distance to stationarity drops from near 1 to near 0 in a window of order n centred at [2(1-\beta)]^{-1} n \log n. For the critical case (\beta =1), we prove that the mixing time is of the order n^{3/2}. In the low-temperature case (\beta > 1), we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n \log n). This is joint work with David Levin and Yuval Peres.