= Ferromagnetic Ising lattice inequality
{title2=$H(\sigma\vee\tau)+H(\sigma\wedge\tau)-H(\sigma)-H(\tau)\ge0$}
For $H(\sigma)=\sum_{xy}J_{xy}\sigma_x\sigma_y+\sum_xh_x\sigma_x$ on a finite graph, assume $J_{xy}\ge0$. Let $U=\{x:\sigma_x=1,\tau_x=-1\}$ and $W=\{x:\sigma_x=-1,\tau_x=1\}$. The displayed difference is $4\sum_{xy:\,x\in U,y\in W\text{ or conversely}}J_{xy}$: all other edge terms and all field terms cancel. Exponentiating proves the <FKG lattice condition> and hence <positive association of random variables> for this <Ising model>.
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