Probabilistic Intuition behind connected correlations and 1PI vertex function

Question

In the context of statistical field or quantum field theory, one encounters so called generating function(al) for connected correlations, aka the following function(al):

$$ W(J) = \ln (Z(J))$$

$$ Z(J) = \int \mathcal{D} \phi e^{-S[\phi]}$$

From a probabilistic standpoint, Z(J) is just the moment generating function(al) while $\ln Z$ is the cumulant generating function(al), I think one generates $\langle \phi^2 \rangle$ for example and the other generates $\langle (\phi - \langle \phi \rangle)^2 \rangle$.

However, we further define another generating function(al), call $\Gamma(\phi)$ usually called the "Effective Action" defined via legendre transform of the function(al) $W(J)$.

1. Why is $\Gamma[\phi]$ the effective action a useful quantity to calculate?
2. What is the probabilistic interpretation of the 1PI diagrams it generates? (cumulant, variance etc...).

3. If correlation functions tell me how lab measurements of $\phi(x)$ and $\phi(y)$ at space/time points x/y would correlate, what do the 1PI functions generated by $\Gamma$ physically represent?

4. Bonus question! The convex nature of the functional W(J) (the fact the Hessian $${\delta^2 \over \delta J(x) \delta J(y)} W(J) > 0 \forall x, y$$) ensures that the legendre transform is bijective. How do we motive this fact physically? Are there cases when this is not true and the effective action is no longer a useful quantity?

Any discussion of the intuition about the quantity $\Gamma[\phi]$ would help, even if it is in the lower dimensional case where $\phi$ is not a field but a single random variable for example.

Answer 1

In probability theory $W=\log Z$ is called the generating function for cumulants. Its Legendre-Fenchel transform $\Gamma$ is the rate function appearing in theorems on large deviations such as Cramer's Theorem. I think if you want more information about the probabilistic meaning of $\Gamma$, then you should lookup the literature on the theory of large deviations. A classic book on the subject is "Large Deviations Techniques and Applications" by Dembo and Zeitouni. A nice introduction with emphasis on statistical mechanics is "A Course on Large Deviations with an Introduction to Gibbs Measures" by Rassoul-Agha and Seppäläinen.

Answer 2

Replace in your formula $S,W,\Gamma$ by the same letters divided by the Boltzmann constant $k$ (the probabilistic analogue of Planck's constant $\hbar$). Then, in the limit of $k\to 0$, the effective action becomes the original action, $\Gamma=S+O( k)$. Thus the effective action is the low temperature (in statistical mechanics ) resp. semiclassical (in quantum field theory) analogue of the (zero temperature, resp. classical) action, and can be determined by an expansion in terms of $k$. As it contains all information about the probability distribution, it is a very useful quantity.