Multiscale Coupling and the Maximum of P(ϕ)2 Models on the Torus

We establish a coupling between the 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 ϕ24 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 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 P(ϕ)2 field on the discretised torus with mesh-size ϵ>0 converges in distribution to a randomly shifted Gumbel distribution as ϵ→0.