High-temperature fast mixing of Glauber dynamics

Let $G$ be a graph on $n$ vertices with maximum degree $\Delta$. The Ising model at inverse temperature $\beta>0$ puts spins $\pm1$ on the vertices with Gibbs distribution $\pi(\sigma)\propto\exp\bigl(\beta\sum_{\{v,w\}\in E}\sigma(v)\sigma(w)\bigr)$, and its Glauber dynamics picks a uniform vertex and re-samples its spin from $\pi$ conditioned on the other spins. For a tolerance $\varepsilon$, the mixing time $t_{\mathrm{mix}}(\varepsilon)$ is the first $t$ with $\max_\sigma\|P^t(\sigma,\cdot)-\pi\|_{TV}\le\varepsilon$, where $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ is the total variation distance.

The theorem (Theorem 15.1 of Levin–Peres–Wilmer, the capstone of Chapter 15) asserts, for every $0<\varepsilon<1$:

1. if $\Delta\tanh\beta<1$, then $\displaystyle t_{\mathrm{mix}}(\varepsilon)\le\Bigl\lceil\frac{n\bigl(\log n+\log(1/\varepsilon)\bigr)}{1-\Delta\tanh\beta}\Bigr\rceil$;
2. if every vertex of $G$ has even degree and $(\Delta/2)\tanh(2\beta)<1$, the same bound holds with $1-(\Delta/2)\tanh(2\beta)$ in the denominator — a strictly weaker temperature condition.

At high temperature the Glauber dynamics mixes in order $n\log n$ steps on any graph — the fundamental fast-mixing criterion for spin systems. The proof is path coupling (Mission VIII) with the one-site coupling whose disagreement probability the tanh lemma controls; since $\tanh\beta<\beta$, condition 1 holds in particular whenever $\beta<1/\Delta$.