Apologies if this comes off as too trivial. The question stems from my reading of Papadodimas and Raju's paper, 'An Infalling Observer in AdS/CFT' (hep-th/1211.6767), in which they introduce the notion of a 'generalized free field'. The n-point functions of such a field may be written as the sum over all possible pairings of n objects, with each pair corresponding to a 2-point function, up to leading order in some parameter 1/N (unrelated to the usual usage of N as the number of flavors).
I was thus led to wonder whether it is possible to have distributions other than the Guassian in, say, a 0-dimensional QFT, which also satisfy Wick's theorem. Additionally, are there distributions (perhaps with the square in the exponent of the Gaussian replaced by an mth power) for which the n-point functions are given by a sum over all possible m-groupings of n objects, so that perturbation theory around such backgrounds would involve dealing with Feynman hypergraphs?
I suppose a combinatorial proof of Wick's theorem can help clarify how far we can go in this direction, but unfortunately, I haven't been able to come up with one.