Cumulant expansion of a coarse-grained free energy
= 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> $F_{\mathrm{eff}}=F_0+\langle V\rangle-\tfrac12(\langle V^2\rangle-\langle V\rangle^2)+\cdots$. The second cumulant selects <connected Feynman diagrams>.