One particle irreducible Feynman diagrams

Section 5 of Borcherds, Barnard, Lectures on Quantum Field Theory is a discussion of the 0-dimensional spacetime case, which gives finite dimensional integrals.

It is important to distinguish giving a sense to all of this nonperturbatively, i.e., for honest functions of $j$ or $\phi$ versus as formal power series in $j$ or $\phi$. In the nonperturbative context, one should use the Fenchel definition of the transform and has to worry about functions being convex. Since there is talk of 1PI diagrams here, the implicit assumption is this is all about formal power series. In that case the way to understand this is as a very special case of the reversion of power series.

Namely, suppose you have $n$ formal power series $F_1,\ldots,F_n$ in $\mathbb{C}[[x_1,\ldots,x_n]]$. Suppose the the constant terms are zero and the linear term of $F_i$ is $\sum_j a_{i\ell}x_\ell$ where the matrix $A=(a_{i\ell})$ is invertible. Then the system of equations $$ \left\{ \begin{array}{c} y_1=F_1(x_1,\ldots,x_n) \\ \vdots \\ y_n=F_n(x_1,\ldots,x_n) \end{array} \right. $$ has a unique solution, i.e., $x_i$'s in $\mathbb{C}[[y_1,\ldots,y_n]]$ without constant terms. Moreover, one has an explicit formula for this solution as a sum over rooted trees where the leaves correspond to the $y$ variables. See my article "Feynman Diagrams in Algebraic Combinatorics" in SLC 2003 where this explained in detail.

Just apply this to the case where $x=j$, $y=\phi$ and $F$ is the gradient of $W$. One finally, needs to insert the almost tautological expansion of $W$ as a sum over trees, with vertices corresponding to 1PI diagrams, and use some simple cancellations.

The "baby version" is actually the regularization of the path integral on a finite Euclidean lattice, which is what is studied by theoretical physicists working in lattice quantum field theory. The usual results about the effective action do indeed hold in that context.