Abstract

We investigate the infinite volume limit of the variational description of Euclidean quantum fields introduced in a previous work. Focusing on two-dimensional theories for simplicity, we prove in detail 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 $\exp <!--FUNCTION\; APPLICATION-->{(\beta \phi )}_2$ 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 $P{(\phi )}_2$ theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations.

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