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This is a second part of my previous question, 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), 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?

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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

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  • $\begingroup$ Carlo, very nice! So, if the pressure converges within a given radius of convergence when the cutoffs are removed, one can use this series as the meaning of the pressure in the infinite volume system? And since this also implies convergence for correlations, the theory is solved. Is this correct? $\endgroup$
    – MathMath
    Commented Aug 9, 2022 at 3:08
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    $\begingroup$ Indeed, that is my understanding. $\endgroup$
    – Carlo Beenakker
    Commented Aug 9, 2022 at 5:58
  • $\begingroup$ Carlo, out of curiosity: do you see any particular reason why the Rivasseau talks about the pressure? He normalizes with the size of the lattice, but this does not seem standard in qft. $\endgroup$
    – MathMath
    Commented Aug 10, 2022 at 12:15
  • $\begingroup$ The reason that eq. 1.7 of Rivasseau is the “pressure” is explained at physics.stackexchange.com/questions/674876/… $\endgroup$
    – Carlo Beenakker
    Commented Aug 10, 2022 at 15:02

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