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

TL;DR

This paper constructs a multiscale coupling between the $ obreak ext{P}( ext{phi})_2$ measure and the Gaussian free field on the torus, enabling analysis of the field's maximum and its convergence to a Gumbel distribution.

Contribution

It introduces a novel coupling method linking $ ext{P}( ext{phi})_2$ measures with Gaussian free fields across scales using the Polchinski renormalisation group and stochastic control techniques.

Findings

Established a multiscale coupling between $ ext{P}( ext{phi})_2$ and Gaussian free field.

Proved the maximum of the $ ext{P}( ext{phi})_2$ field converges to a Gumbel distribution.

Applied regularity estimates to analyze the field's maximum behavior.

Abstract

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 $\phi^4_2$ measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Bou\'e-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 $\epsilon \rightarrow 0$.