# What are the tempered Gibbs measures of classical $\phi^4$-theory?

I consider classical $$\phi^4$$-theory on the lattice. The model is defined in finite volume with Hamiltonian

\begin{align*} H(\phi) = - \sum_{x \sim y} J_{x,y} \phi_x \phi_y \end{align*} and a-priori measure \begin{align*} \rho(d \phi_x) = e^{- \lambda \phi_x^4 + b \phi_x^2} d\phi_x. \end{align*} Here the fields can be in $$\mathbb{R}$$, $$\lambda$$ is positive and I am fine with considering only ferromagnetic interactions of the form $$J_{x,y} = J > 0$$.

This model fits into the framework of Lebowitz and Presutti,Statistical mechanics of unbounded spins CMP. 50, 195—218 (1976). Which means that I can define the class of tempered (infinite volume) Gibbs measures for this particular model. That means that I get $$\mu_{\pm}$$ defined as limits of states on $$\Lambda_R$$ with boundary conditions $$\pm a \log(R)$$ for an $$a >0$$ with all the tempered Gibbs measures lying in between $$\mu_-$$ and $$\mu_+$$.

However, as the paper is in larger generality and the $$\phi^4$$-model itself is very well studied. So I wonder what additional results can be shown for this particular model. In particular I wonder

1. Whether all translation invariant Gibbs measures are tempered?
2. What happens if I choose boundary conditions $$\pm a R$$ instead of $$\pm a \log(R)$$
3. In what sense is the class of tempered Gibbs measures the class that is natural to study and what classifications theorems (forming a Choquet simplex etc.) are know about tempered translation invariant Gibbs states and just translation invariant Gibbs states?

However, the litterature of the model is also a wilderness so I hope you can help me out.