In QFT, either at zero or finite temperature, the Linked Cluster Theorem (LCT) ensures that all disconnected diagrams appearing in the numerator of the interacting Green's function exactly cancel with the denominator, which evaluates the $S$-matrix expansion. This way, one only needs to care about evaluating connected diagrams in the perturbative expansion, and Dyson's equation includes only self-energy terms composed of connected graphs.
The question is how general can one expect this to happen for other correlation functions. For instance, for correlations involving density fluctuations, is there a proof that the LCT still holds? I might be wrong, but it is not immediate that for general correlators such cancellations should take place, given that the evaluation of $\text{Tr}\left(e^{-\beta(H-\mu N)} S(\beta)\right)$ (the denominator of the expansion at finite temperature) is unique, whereas the possible Wick contractions for different correlators is not.
