I was reading Constructive Renormalization Group by V. Rivasseau and I got some points which I would like to clarify.
On page 2, Rivasseau talks about the large field problem and, if I understood it correctly, he says that the problem arises when one tries to study integrals with respect to the measure: $$d\mu_{C_{\kappa}}(\phi) = e^{-\lambda \int_{\Lambda}\phi^{4}(x)dx}$$ by Taylor expanding the exponential of $e^{-\lambda \int_{\Lambda}\phi^{4}(x)dx}$. When one expands it, one gets a series which is divergent and, at most, Borel summable, although the integral itself can be finite.
The point of the text, however, is to discuss nonperturbative RG. On the section "Constructive RG is necessary!" he states that the large field problem can arise when performing the RG transformation already in the second step of the iteraction and that "the behavior of $S_{\text{eff}}(\phi)$ at large $\phi$ is unclear, so that starting from a stable interaction, even the second step of the RG may be already ill-defined." From this sentence I understand that one can start if a given stable interaction and at the second step the effective action is no longer well-defined, but I can't understand why this is the case even when IR-cutoffs are being consider. Also, what does he mean by "large $\phi$"? $\phi$ is a function of $\mathscr{S}(\mathbb{R}^{d})$, so what does it mean $\phi$ to be large? I would be glad to see a simple example ilustrating what is going on here, if possible. I mean, a stable bare action becoming "large" after a single step of the RG.
