Display (1), §3.2, p. 16 — χ²(π_{i+1}, π_i) = Z(β_i)Z(2β_{i+1} − β_i)/Z(β_{i+1})² − 1

Let $\Omega$ be finite, $H:\Omega\to\mathbb N$ a Hamiltonian with at least one configuration of energy $0$, and $0=\beta_0<\beta_1<\dots<\beta_\ell=\infty$ a cooling schedule, with Gibbs distributions $\pi_i(x)=e^{-\beta_iH(x)}/Z(\beta_i)$. Then for every $i\in\{0,\dots,\ell-1\}$,

The identity includes the last step $i=\ell-1$, where $\beta_{i+1}=\infty$ and $2\beta_{i+1}-\beta_i=\infty$. It shows that the Chebyshev cooling condition of Definition 1 is exactly a bound on $\chi^2(\pi_{i+1},\pi_i)+1$.

Formalization Note The hypothesis that some $x$ has $H(x)=0$ makes $Z(\infty)>0$, without which $\pi_\ell$ is undefined; it is added and disclosed. The bound $H\le n$ of the paper's model is not needed for this identity and is omitted.