The tanh contraction lemma

Fix an inverse temperature $\beta>0$ and consider the function

which measures how much the conditional probability of a $+$ spin under the Ising Glauber dynamics moves when the neighbourhood field increases from $x-1$ to $x+1$ (the heat-bath update at a site with neighbouring spin sum $S$ chooses $+$ with probability $(1+\tanh(\beta S))/2$).

The theorem (Lemma 15.2 of Levin–Peres–Wilmer) asserts four elementary properties:

1. $g$ is even: $g(-x)=g(x)$ for every real $x$;
2. $g$ is non-increasing on $[0,\infty)$: $0\le x\le y$ implies $g(y)\le g(x)$;
3. $g(x)\le 2\tanh\beta$ for every real $x$ — the value at $x=0$ is the global maximum;
4. over odd integers the sharper bound holds: $g(k)\le\tanh(2\beta)$ for every odd $k\in\mathbb Z$ — the value at $k=\pm1$ is the maximum among odd integers.

These four inequalities are the entire analytic content of the high-temperature theorem: in the one-site coupling of two adjacent configurations, the probability that the updated site disagrees is controlled by $g$ evaluated at the neighbouring spin sum, which for a graph with all degrees even is an odd integer — whence the two conditions $\Delta\tanh\beta<1$ and $(\Delta/2)\tanh(2\beta)<1$ of the goal theorem.