Cumulant expansion of a coarse-grained free energy

Writing the interaction as $V$ and averaging over the fast modes, which have a Gaussian distribution, gives the cumulant expansion $F_{\mathrm{eff}}=F_0+⟨V⟩-\frac{1}{2}(⟨V^2⟩-⟨V{⟩}^2)+\cdots$. The second cumulant selects connected Feynman diagrams.