= Ferromagnetic Ising lattice inequality
{title2=$H(\sigma\vee\tau)+H(\sigma\wedge\tau)-H(\sigma)-H(\tau)\ge0$}
Let $H(\sigma)=\sum_{\{x,y\}}J_{xy}\sigma_x\sigma_y+\sum_xh_x\sigma_x$ on a finite graph, with each undirected edge counted once, $J_{xy}\ge0$, and <Ising spins> $\sigma_x\in\{-1,1\}$. Here $H$ is a dimensionless log-weight: the <probability mass function> is $\pi(\sigma)=e^{H(\sigma)}/Z$. If $E$ denotes physical <energy>, the convention is $H=-\beta E$, not $H=E$.
Order configurations coordinatewise, so $(\sigma\vee\tau)_x=\max(\sigma_x,\tau_x)$ and $(\sigma\wedge\tau)_x=\min(\sigma_x,\tau_x)$. Define $U=\{x:\sigma_x=1,\tau_x=-1\}$ and $W=\{x:\sigma_x=-1,\tau_x=1\}$. An edge contributes zero to the displayed difference unless it joins $U$ to $W$. On such an edge the joined and met configurations each have <spin> product $+1$, whereas both original configurations have product $-1$. Its contribution is therefore $4J_{xy}$. Each field term cancels because the coordinatewise maximum and minimum preserve the sum of the two <spins>. Consequently,
$$
H(\sigma\vee\tau)+H(\sigma\wedge\tau)-H(\sigma)-H(\tau)=4\sum_{\{x,y\}:\,x\in U,\,y\in W\text{ or conversely}}J_{xy}\ge0.
$$
Exponentiating gives $\pi(\sigma\vee\tau)\pi(\sigma\wedge\tau)\ge\pi(\sigma)\pi(\tau)$, the <FKG lattice condition>. The corresponding <FKG inequality> implies <positive association of random variables>: increasing observables have nonnegative <covariance>. The sign convention matters: using a positive ferromagnetic log-weight as though it were physical <energy> in $e^{-\beta E}$ would reverse which couplings favor alignment.
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