Multiscale Coupling and the Maximum of $${\mathcal {P}}(\phi )_2$$ Models on the Torus

Abstract

We establish a coupling between the $${\mathcal {P}}(\phi )_2$$ P ( ϕ ) 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 $$\phi ^4_2$$ ϕ 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 $${\mathcal {P}}(\phi )_2$$ P ( ϕ ) 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 $${\mathcal {P}}(\phi )_2$$ P ( ϕ ) 2 field on the discretised torus with mesh-size $$\epsilon > 0$$ ϵ > 0 converges in distribution to a randomly shifted Gumbel distribution as $$\epsilon \rightarrow 0$$ ϵ → 0 .