Curie--Weiss: fast mixing for $\alpha<1$

The Curie–Weiss model is the Ising model on the complete graph $K_n$: spins $\pm1$ on $n$ vertices, every pair interacting, with Gibbs distribution $\pi(\sigma)\propto\exp\bigl(\beta\sum_{\{v,w\}}\sigma(v)\sigma(w)\bigr)$ at inverse temperature $\beta=\alpha/n$ — the $1/n$ scaling that makes the total interaction per site of constant order, with $\alpha$ the effective temperature parameter. The Glauber dynamics re-samples a uniformly chosen site from the conditional distribution; the mixing time $t_{\mathrm{mix}}(\varepsilon)$ is the first $t$ with $\max_\sigma\|P^t(\sigma,\cdot)-\pi\|_{TV}\le\varepsilon$, where $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$.

The theorem (Theorem 15.3(i) of Levin–Peres–Wilmer) asserts: for every $n\ge2$, every $0<\alpha<1$, and every $0<\varepsilon<1$,

Below the critical value $\alpha=1$ the mean-field dynamics mixes in order $n\log n$ steps. The proof is one line from the high-temperature theorem of this mission: on $K_n$ the degree is $n-1$ and $(n-1)\tanh(\alpha/n)\le\alpha$. The companion theorem shows that above $\alpha=1$ the same dynamics needs exponentially many steps — the dynamical phase transition.