Curie--Weiss: exponentially slow mixing for $\alpha>1$

The Curie–Weiss model is the Ising model on the complete graph $K_n$: spins $\pm1$ on $n$ vertices with Gibbs distribution $\pi(\sigma)\propto\exp\bigl(\beta\sum_{\{v,w\}}\sigma(v)\sigma(w)\bigr)$ at $\beta=\alpha/n$, and the Glauber dynamics re-samples a uniformly chosen site from the conditional distribution. The mixing time $t_{\mathrm{mix}}$ is the first $t$ at which $\max_\sigma\|P^t(\sigma,\cdot)-\pi\|_{TV}\le\tfrac14$, with $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ the total variation distance.

The theorem (Theorem 15.3(ii) of Levin–Peres–Wilmer) asserts: for every $\alpha>1$ there are a rate $r>0$ and a constant $C>0$ such that for all sufficiently large $n$,

Above the critical temperature parameter the dynamics is exponentially slow — the low-temperature half of the dynamical phase transition, in the sharpest possible contrast with the $n\log n$ mixing below $\alpha=1$. The obstruction is an energy barrier: at $\alpha>1$ the magnetization $\sum_v\sigma(v)$ concentrates near two symmetric values $\pm m^*n$, and passing from one well to the other forces the chain through configurations of exponentially small stationary mass; the bottleneck bound of Mission IV converts that barrier into the exponential lower bound.