On the variational method for Euclidean quantum fields in infinite volume

We investigate the infinite volume limit of the variational description of Euclidean quantum fields introduced in a previous work. Focussing on two dimensional theories for simplicity, we prove in details how to use the variational approach to obtain tightness of ${\phi }_2^4$ without cutoffs and a corresponding large deviation principle for any infinite volume limit. Any infinite volume measure is described via a forward–backwards stochastic differential equation in weak form (wFBSDE). Similar considerations apply to more general $P(\phi {)}_2$ theories. We consider also the $\exp (\beta \varphi)_2$modelfor$\beta^2 < 8 \pi$(thesocalledfull$L^1$ regime) and prove uniqueness of the infinite volume limit and a variational characterization of the unique infinite volume measure. The corresponding characterization for $P(\phi {)}_2$ theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations.