  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 $\varphi^4_2$ 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 $P (\varphi)_2$ theories. We consider also the $\exp (\beta
\varphi)_2$ model for $\beta^2 < 8 \pi$ (the so called full $L^1$ regime) and
prove uniqueness of the infinite volume limit and a variational
characterization of the unique infinite volume measure. The corresponding
characterization for $P (\varphi)_2$ theories is lacking due to the difficulty
of studying the stability of the wFBSDE against local perturbations.
