Multiscale coupling and the maximum of $\mathcal{P}(\phi {)}_2$ models on the torus

We establish a coupling between the $\mathcal{P}(\phi {)}_2$ measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied ${\phi }_2^4$ measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué-Dupuis stochastic control representation for $\mathcal{P}(\phi {)}_2$. More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field. As an application of the coupling, we prove that the maximum of the $\mathcal{P}(\phi {)}_2$ field on the discretised torus with mesh-size $\epsilon >0$ converges in distribution to a randomly shifted Gumbel distribution as $\epsilon \to 0$.