Ising on the cycle: $n\log n$ mixing at every temperature

The Ising model on the $n$-cycle puts spins $\pm1$ on $\mathbb Z_n$ (each residue adjacent to its two neighbours) with Gibbs distribution $\pi(\sigma)\propto\exp\bigl(\beta\sum_{\{v,w\}\in E}\sigma(v)\sigma(w)\bigr)$ at inverse temperature $\beta>0$, and its Glauber dynamics re-samples a uniformly chosen site from the conditional distribution. 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 $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$. Set

which is positive for every $\beta$.

The theorem (Theorem 15.4 of Levin–Peres–Wilmer) asserts: for any fixed $0<\varepsilon<1$ and any margin $\delta>0$ there is an $N$ such that for all $n\ge N$,

On the cycle the dynamics mixes in $n\log n$ steps at every temperature — no phase transition in one dimension, in sharp contrast to the complete graph of the companion theorems. The upper bound is the even-degree case of the high-temperature theorem (every vertex of the cycle has degree $2$, so the condition $(\Delta/2)\tanh(2\beta)=\tanh(2\beta)<1$ always holds); the lower bound runs Wilson's method (Mission VII) with a Fourier-mode eigenfunction.