On the variational method for Euclidean quantum fields in infinite volume

Understand

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.