This is a second part of my [previous question](https://mathoverflow.net/questions/428017/what-is-a-large-field-problem), which I decided to split into two parts not to mix up different topics at one giant question.


Again, according to V. Rivasseau (section 1.5 of [Constructive Renormalization Theory](https://arxiv.org/pdf/math-ph/9902023.pdf)), the solution for the large field problem is to consider a single scale analysis of the RG. The text explains that the main tool of constructive RG is to do a cluster expansion and then a Mayer expansion. The cluster expansion is related to the partition function of the theory and the Mayer expansion is related to its logarithm, which is the pressure.


My first question concerns the terminology: why is it called *constructive*? And if one *expands* the partition function and the pressure in (what I understand is the) power series of the coupling parameter $\lambda$, why isn't it considered a *perturbation* theory?


My second question concerns the objective of the theory. If I understood it correctly, the point is to expand the pressure in power series of $\lambda$ and prove it is uniformly bounded, so it has nonzero radius of convergence when the cutoffs are removed. What information can we get from that? Does it imply that perturbation theory converges? And if so, why is that? Isn't the whole idea to obtain the Schwinger (or $n$ point functions)? Can one obtain such functions from the pressure?



Cluster expansion, Mayer expansion and perturbative renormalization group
I think there is a misunderstanding here on what is the expansion parameter.


Perturbative renormalization expands in a power series of the interaction strength (the coupling parameter $\lambda$); this expansion typically has zero radius of convergence.


Constructive renormalization, instead, does not expand in powers of $\lambda$, but in the number of particles that interact. The interaction strength is not expanded, it is retained to all orders in $\lambda$.


The objective of the theory is to show that this expansion (known as a cluster or Mayer expansion) has a finite radius of convergence. This is typically the case if the interaction decays sufficiently rapidly at large distances.


And yes, convergence of the pressure implies convergence of correlators, see for example <https://doi.org/10.1063/1.523040>


